Journal of Differential Geometry Articles (Project Euclid)
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The latest articles from Journal of Differential Geometry on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTWed, 04 May 2011 09:16 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Classification of compact ancient solutions to the curve shortening flow
http://projecteuclid.org/euclid.jdg/1279114297
<strong>Panagiota Daskalopoulos</strong>, <strong>Richard Hamilton</strong>, <strong>Natasa Sesum</strong><p><strong>Source: </strong>J. Differential Geom., Volume 84, Number 3, 455--464.</p>projecteuclid.org/euclid.jdg/1279114297_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTA sub-Riemannian Santaló formula with applications to isoperimetric inequalities and first Dirichlet eigenvalue of hypoelliptic operatorshttps://projecteuclid.org/euclid.jdg/1549422105<strong>Dario Prandi</strong>, <strong>Luca Rizzi</strong>, <strong>Marcello Seri</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 2, 339--379.</p><p><strong>Abstract:</strong><br/>
In this paper we prove a sub-Riemannian version of the classical Santaló formula: a result in integral geometry that describes the intrinsic Liouville measure on the unit cotangent bundle in terms of the geodesic flow. Our construction works under quite general assumptions, satisfied by any sub-Riemannian structure associated with a Riemannian foliation with totally geodesic leaves (e.g., CR and QC manifolds with symmetries), any Carnot group, and some non-equiregular structures such as the Martinet one. A key ingredient is a “reduction procedure” that allows to consider only a simple subset of sub-Riemannian geodesics.
As an application, we derive isoperimetric-type and ($p$-)Hardy-type inequalities for a compact domain $M$ with piecewise $C^{1,1}$ boundary, and a universal lower bound for the first Dirichlet eigenvalue $\lambda_1 (M)$ of the sub-Laplacian,
\[ \lambda_1 (M) \geq \frac{k \pi^2}{L^2} \; \textrm{,} \]
in terms of the rank $k$ of the distribution and the length $L$ of the longest reduced sub-Riemannian geodesic contained in $M$. All our results are sharp for the sub-Riemannian structures on the hemispheres of the complex and quaternionic Hopf fibrations:
\[ \mathbb{S}^1 \hookrightarrow \mathbb{S}^{2d+1} \overset{p}{\to} \mathbb{CP}^d \; \textrm{,} \qquad \mathbb{S}^3 \hookrightarrow \mathbb{S}^{4d+3} \overset{p}{\to} \mathbb{HP}^d \; \textrm{,} \qquad {d \geq 1} \; \textrm{,} \]
where the sub-Laplacian is the standard hypoelliptic operator of CR and QC geometries, $L = \pi$ and $k = 2d$ or $4d$, respectively.
</p>projecteuclid.org/euclid.jdg/1549422105_20190205220216Tue, 05 Feb 2019 22:02 ESTUnique asymptotics of ancient convex mean curvature flow solutionshttps://projecteuclid.org/euclid.jdg/1552442605<strong>Sigurd Angenent</strong>, <strong>Panagiota Daskalopoulos</strong>, <strong>Natasa Sesum</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 3, 381--455.</p><p><strong>Abstract:</strong><br/>
We study compact noncollapsed ancient convex solutions to Mean Curvature Flow in $\mathbb{R}^{n+1}$ with $O(1) \times O(n)$ symmetry. We show they all have unique asymptotics as $t \to -\infty$ and we give a precise asymptotic description of these solutions. The asymptotics apply, in particular, to the solutions constructed by White, and Haslhofer and Hershkovits (in the case of those particular solutions the asymptotics were predicted and formally computed by Angenent).
</p>projecteuclid.org/euclid.jdg/1552442605_20190312220339Tue, 12 Mar 2019 22:03 EDTImmersing quasi-Fuchsian surfaces of odd Euler characteristic in closed hyperbolic $3$-manifoldshttps://projecteuclid.org/euclid.jdg/1552442607<strong>Yi Liu</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 3, 457--493.</p><p><strong>Abstract:</strong><br/>
In this paper, it is shown that every closed hyperbolic $3$-manifold contains an immersed quasi-Fuchsian closed subsurface of odd Euler characteristic. The construction adopts the good pants method, and the primary new ingredient is an enhanced version of the connection principle, which allows one to connect any two frames with a path of frames in a prescribed relative homology class of the frame bundle. The existence result is applied to show that every uniform lattice of $\mathrm{PSL}(2, \mathbb{C})$ admits an exhausting nested sequence of sublattices with exponential homological torsion growth. However, the constructed sublattices are not normal in general.
</p>projecteuclid.org/euclid.jdg/1552442607_20190312220339Tue, 12 Mar 2019 22:03 EDTStability of torsion-free $\mathrm{G}_2$ structures along the Laplacian flowhttps://projecteuclid.org/euclid.jdg/1552442608<strong>Jason D. Lotay</strong>, <strong>Yong Wei</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 3, 495--526.</p><p><strong>Abstract:</strong><br/>
We prove that torsion-free $\mathrm{G}_2$ structures are (weakly) dynamically stable along the Laplacian flow for closed $\mathrm{G}_2$ structures. More precisely, given a torsion-free $\mathrm{G}_2$ structure $\overline{\varphi}$ on a compact $7$-manifold $M$, the Laplacian flow with initial value in $[\overline{\varphi}]$, sufficiently close to $\overline{\varphi}$, will converge to a point in the $\mathrm{Diff}^0 (M)$-orbit of $\overline{\varphi}$. We deduce, from fundamental work of Joyce, that the Laplacian flow starting at any closed $\mathrm{G}_2$ structure with sufficiently small torsion will exist for all time and converge to a torsion-free $\mathrm{G}_2$ structure.
</p>projecteuclid.org/euclid.jdg/1552442608_20190312220339Tue, 12 Mar 2019 22:03 EDTDecorated super-Teichmüller spacehttps://projecteuclid.org/euclid.jdg/1552442609<strong>R. C. Penner</strong>, <strong>Anton M. Zeitlin</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 3, 527--566.</p><p><strong>Abstract:</strong><br/>
We introduce coordinates for a principal bundle $S\tilde{T}(F)$ over the super Teichmüller space $ST(F)$ of a surface F with $s \geq 1$ punctures that extend the lambda length coordinates on the decorated bundle $\tilde{T}(F) = T(F) \times \mathbb{R}^s_{+}$ over the usual Teichmüller space $T(F)$. In effect, the action of a Fuchsian subgroup of $PSL (2, \mathbb{R})$ on Minkowski space $\mathbb{R}^{2,1}$ is replaced by the action of a super Fuchsian subgroup of $OSp (1\vert 2)$ on the super Minkowski space $\mathbb{R}^{2, 1 \vert 2}$, where $OSp (1\vert 2)$ denotes the orthosymplectic Lie supergroup, and the lambda lengths are extended by fermionic invariants of suitable triples of isotropic vectors in $\mathbb{R}^{2, 1 \vert 2}$. As in the bosonic case, there is the analogue of the Ptolemy transformation now on both even and odd coordinates as well as an invariant even two-form on $S\tilde{T}(F)$ generalizing the Weil–Petersson Kähler form. This, finally, solves a problem posed in Yuri Ivanovitch Manin’s Moscow seminar some thirty years ago to find the super analogue of decorated Teichmüller theory and provides a natural geometric interpretation in $\mathbb{R}^{2, 1 \vert 2}$ for the super moduli of $S\tilde{T}(F)$.
</p>projecteuclid.org/euclid.jdg/1552442609_20190312220339Tue, 12 Mar 2019 22:03 EDTKohn–Rossi cohomology and nonexistence of CR morphisms between compact strongly pseudoconvex CR manifoldshttps://projecteuclid.org/euclid.jdg/1552442610<strong>Stephen S.-T. Yau</strong>, <strong>Huaiqing Zuo</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 3, 567--580.</p><p><strong>Abstract:</strong><br/>
One of the fundamental questions in CR geometry is: Given two strongly pseudoconvex CR manifolds $X_1$ and $X_2$ of dimension $2n-1$, is there a non-constant CR morphism between them? In this paper, we use Kohn–Rossi cohomology to show the non-existence of non-constant CR morphism between such two CR manifolds. Specifically, if $\dim H^{p,q}_{KR} (X_1) \lt \dim H^{p,q}_{KR} (X_2)$ for any $(p, q)$ with $1 \leq q \leq n-2$, then there is no non-constant CR morphism from $X_1$ to $X_2$.
</p>projecteuclid.org/euclid.jdg/1552442610_20190312220339Tue, 12 Mar 2019 22:03 EDTIndex to Volume 111https://projecteuclid.org/euclid.jdg/1552442611<p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 3</p>projecteuclid.org/euclid.jdg/1552442611_20190312220339Tue, 12 Mar 2019 22:03 EDTLagrangian cobordism and metric invariantshttps://projecteuclid.org/euclid.jdg/1557281005<strong>Octav Cornea</strong>, <strong>Egor Shelukhin</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 112, Number 1, 1--45.</p><p><strong>Abstract:</strong><br/>
We introduce new pseudo-metrics on spaces of Lagrangian submanifolds of a symplectic manifold $(M, \omega)$ by considering areas associated to projecting Lagrangian cobordisms in $\mathbb{C} \times M$ to the “time-energy plane” $\mathbb{C}$. We investigate the non-degeneracy properties of these pseudo-metrics, reflecting the rigidity and flexibility aspects of Lagrangian cobordisms.
</p>projecteuclid.org/euclid.jdg/1557281005_20190507220348Tue, 07 May 2019 22:03 EDTMinimal surfaces for Hitchin representationshttps://projecteuclid.org/euclid.jdg/1557281006<strong>Song Dai</strong>, <strong>Qiongling Li</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 112, Number 1, 47--77.</p><p><strong>Abstract:</strong><br/>
Given a reductive representation $\rho : \pi_1 (S) \to G$, there exists a $\rho$-equivariant harmonic map $f$ from the universal cover of a fixed Riemann surface $\Sigma$ to the symmetric space $G/K$ associated to $G$. If the Hopf differential of $f$ vanishes, the harmonic map is then minimal. In this paper, we investigate the properties of immersed minimal surfaces inside symmetric space associated to a subloci of Hitchin component: the $q_n$ and $q_{n-1}$ cases. First, we show that the pullback metric of the minimal surface dominates a constant multiple of the hyperbolic metric in the same conformal class and has a strong rigidity property. Secondly, we show that the immersed minimal surface is never tangential to any flat inside the symmetric space. As a direct corollary, the pullback metric of the minimal surface is always strictly negatively curved. In the end, we find a fully decoupled system to approximate the coupled Hitchin system.
</p>projecteuclid.org/euclid.jdg/1557281006_20190507220348Tue, 07 May 2019 22:03 EDTALF gravitational instantons and collapsing Ricci-flat metrics on the $K3$ surfacehttps://projecteuclid.org/euclid.jdg/1557281007<strong>Lorenzo Foscolo</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 112, Number 1, 79--120.</p><p><strong>Abstract:</strong><br/>
We construct large families of new collapsing hyperkähler metrics on the $K3$ surface. The limit space is a flat Riemannian $3$-orbifold $T^3 / \mathbb{Z}_2$. Away from finitely many exceptional points the collapse occurs with bounded curvature. There are at most $24$ exceptional points where the curvature concentrates, which always contains the 8 fixed points of the involution on $T^3$. The geometry around these points is modelled by ALF gravitational instantons: of dihedral type ($D_k$) for the fixed points of the involution on T3 and of cyclic type ($A_k$) otherwise.
The collapsing metrics are constructed by deforming approximately hyperkähler metrics obtained by gluing ALF gravitational instantons to a background (incomplete) $S^1$–invariant hyperkähler metric arising from the Gibbons–Hawking ansatz over a punctured $3$-torus.
As an immediate application to submanifold geometry, we exhibit hyperkähler metrics on the $K3$ surface that admit a strictly stable minimal sphere which cannot be holomorphic with respect to any complex structure compatible with the metric.
</p>projecteuclid.org/euclid.jdg/1557281007_20190507220348Tue, 07 May 2019 22:03 EDTReal submanifolds of maximum complex tangent space at a CR singular point, IIhttps://projecteuclid.org/euclid.jdg/1557281008<strong>Xianghong Gong</strong>, <strong>Laurent Stolovitch</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 112, Number 1, 121--198.</p><p><strong>Abstract:</strong><br/>
We study germs of real analytic $n$-dimensional submanifold of $\mathbf{C}^n$ that has a complex tangent space of maximal dimension at a CR singularity. Under some assumptions, we first classify holomorphically the quadrics having this property. We then study higher order perturbations of these quadrics and their transformations to a normal form under the action of local (possibly formal) biholomorphisms at the singularity. We are led to study formal Poincaré–Dulac normal forms (non-unique) of reversible biholomorphisms. We exhibit a reversible map of which the normal forms are all divergent at the singularity. We then construct a unique formal normal form of the submanifolds under a non degeneracy condition.
</p>projecteuclid.org/euclid.jdg/1557281008_20190507220348Tue, 07 May 2019 22:03 EDTSymplectic embeddings from concave toric domains into convex oneshttps://projecteuclid.org/euclid.jdg/1559786421<strong>Dan Cristofaro-Gardiner</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 112, Number 2, 199--232.</p><p><strong>Abstract:</strong><br/>
Embedded contact homology gives a sequence of obstructions to four-dimensional symplectic embeddings, called ECH capacities. In “Symplectic embeddings into four-dimensional concave toric domains”, the author, Choi, Frenkel, Hutchings and Ramos computed the ECH capacities of all “concave toric domains”, and showed that these give sharp obstructions in several interesting cases. We show that these obstructions are sharp for all symplectic embeddings of concave toric domains into “convex” ones. In an appendix with Choi, we prove a new formula for the ECH capacities of convex toric domains, which shows that they are determined by the ECH capacities of a corresponding collection of balls.
</p>projecteuclid.org/euclid.jdg/1559786421_20190605220045Wed, 05 Jun 2019 22:00 EDTProperly immersed surfaces in hyperbolic $3$-manifoldshttps://projecteuclid.org/euclid.jdg/1559786424<strong>William H. Meeks</strong>, <strong>Álvaro K. Ramos</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 112, Number 2, 233--261.</p><p><strong>Abstract:</strong><br/>
We study complete finite topology immersed surfaces $\Sigma$ in complete Riemannian $3$-manifolds $N$ with sectional curvature $K_N \leq -a^2 \leq 0$, such that the absolute mean curvature function of $\Sigma$ is bounded from above by a and its injectivity radius function is not bounded away from zero on each of its annular end representatives. We prove that such a surface $\Sigma$ must be proper in $N$ and its total curvature must be equal to $2 \pi \chi (\Sigma)$. If $N$ is a hyperbolic $3$-manifold of finite volume and $\Sigma$ is a properly immersed surface of finite topology with nonnegative constant mean curvature less than $1$, then we prove that each end of $\Sigma$ is asymptotic (with finite positive integer multiplicity) to a totally umbilic annulus, properly embedded in $N$.
</p>projecteuclid.org/euclid.jdg/1559786424_20190605220045Wed, 05 Jun 2019 22:00 EDTRigidity of pairs of rational homogeneous spaces of Picard number $1$ and analytic continuation of geometric substructures on uniruled projective manifoldshttps://projecteuclid.org/euclid.jdg/1559786425<strong>Ngaiming Mok</strong>, <strong>Yunxin Zhang</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 112, Number 2, 263--345.</p><p><strong>Abstract:</strong><br/>
Building on the geometric theory of uniruled projective manifolds by Hwang–Mok, which relies on the study of varieties of minimal rational tangents (VMRTs) from both the algebro-geometric and the differential-geometric perspectives, Mok, Hong–Mok and Hong–Park have studied standard embeddings between rational homogeneous spaces $X = G/P$ of Picard number $1$. Denoting by $S \subset X$ an arbitrary germ of complex submanifold which inherits from $X$ a geometric structure defined by taking intersections of VMRTs with tangent subspaces and modeled on some rational homogeneous space $X_0 = G_0 / P_0$ of Picard number $1$ embedded in $X = G/P$ as a linear section through a standard embedding, we say that $(X_0, X)$ is rigid if there always exists some $\gamma \in \mathrm{Aut}(X)$ such that $S$ is an open subset of $\gamma (X_0)$. We prove that a pair $(X_0, X)$ of sub-diagram type is rigid whenever $X_0$ is nonlinear, which in the Hermitian symmetric case recovers Schubert rigidity for nonlinear smooth Schubert cycles, and which in the general rational homogeneous case goes beyond earlier works dealing with images of holomorphic maps. Our methods apply to uniruled projective manifolds $(X, \mathcal{K})$, for which we introduce a general notion of sub-VMRT structures $\varpi : \mathscr{C} (S) \to S$, proving that they are rationally saturated under an auxiliary condition on the intersection $\mathscr{C} (S) := \mathscr{C} (X) \cap \mathbb{P} T (S)$ and a nondegeneracy condition for substructures expressed in terms of second fundamental forms on VMRTs. Under the additional hypothesis that minimal rational curves are of degree $1$ and that distributions spanned by sub-VMRTs are bracket generating, we prove that $S$ extends to a subvariety $Z \subset X$. For its proof, starting with a “Thickening Lemma” which yields smooth collars around certain standard rational curves, we show that the germ of submanifold $(S; x_0)$ and, hence, the associated germ of sub-VMRT structure on $(S; x_0)$ can be propagated along chains of “thickening” curves issuing from $x_0$, and construct by analytic continuation a projective family of chains of rational curves compactifying the latter family, thereby constructing the projective completion $Z$ of $S$ as its image under the evaluation map.
</p>projecteuclid.org/euclid.jdg/1559786425_20190605220045Wed, 05 Jun 2019 22:00 EDTSharp fundamental gap estimate on convex domains of spherehttps://projecteuclid.org/euclid.jdg/1559786428<strong>Shoo Seto</strong>, <strong>Lili Wang</strong>, <strong>Guofang Wei</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 112, Number 2, 347--389.</p><p><strong>Abstract:</strong><br/>
In their celebrated work, B. Andrews and J. Clutterbuck proved the fundamental gap (the difference between the first two eigenvalues) conjecture for convex domains in the Euclidean space [3] and conjectured similar results hold for spaces with constant sectional curvature. We prove the conjecture for the sphere. Namely when $D$, the diameter of a convex domain in the unit $\mathbb{S}^n$ sphere, is $\leq \frac{\pi}{2}$, the gap is greater than the gap of the corresponding 1-dim sphere model. We also prove the gap is $\geq 3 \frac{\pi^2}{D^2}$ when $n \geq 3$, giving a sharp bound. As in [3], the key is to prove a super log-concavity of the first eigenfunction.
</p>projecteuclid.org/euclid.jdg/1559786428_20190605220045Wed, 05 Jun 2019 22:00 EDTDehn filling and the Thurston normhttps://projecteuclid.org/euclid.jdg/1563242469<strong>Kenneth L. Baker</strong>, <strong>Scott A. Taylor</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 112, Number 3, 391--409.</p><p><strong>Abstract:</strong><br/>
For a compact, orientable, irreducible $3$-manifold with toroidal boundary that is not the product of a torus and an interval or a cable space, each boundary torus has a finite set of slopes such that, if avoided, the Thurston norm of a Dehn filling behaves predictably. More precisely, for all but finitely many slopes, the Thurston norm of a class in the second homology of the filled manifold plus the so-called winding norm of the class will be equal to the Thurston norm of the corresponding class in the second homology of the unfilled manifold. This generalizes a result of Sela and is used to answer a question of Baker–Motegi concerning the Seifert genus of knots obtained by twisting a given initial knot along an unknot which links it.
</p>projecteuclid.org/euclid.jdg/1563242469_20190715220128Mon, 15 Jul 2019 22:01 EDTMin-max embedded geodesic lines in asymptotically conical surfaceshttps://projecteuclid.org/euclid.jdg/1563242470<strong>Alessandro Carlotto</strong>, <strong>Camillo De Lellis</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 112, Number 3, 411--445.</p><p><strong>Abstract:</strong><br/>
We employ min-max methods to construct uncountably many, geometrically distinct, properly embedded geodesic lines in any asymptotically conical surface of non-negative scalar curvature, a setting where minimization schemes are doomed to fail. Our construction provides control of the Morse index of the geodesic lines we produce, which will be always less or equal than one (with equality under suitable curvature or genericity assumptions), as well as of their precise asymptotic behavior. In fact, we can prove that in any such surface for every couple of opposite half-lines there exists an embedded geodesic line whose two ends are asymptotic, in a suitable sense, to those half-lines.
</p>projecteuclid.org/euclid.jdg/1563242470_20190715220128Mon, 15 Jul 2019 22:01 EDTQuantitative flatness results and $BV$-estimates for stable nonlocal minimal surfaceshttps://projecteuclid.org/euclid.jdg/1563242471<strong>Eleonora Cinti</strong>, <strong>Joaquim Serra</strong>, <strong>Enrico Valdinoci</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 112, Number 3, 447--504.</p><p><strong>Abstract:</strong><br/>
We establish quantitative properties of minimizers and stable sets for nonlocal interaction functionals, including the $s$-fractional perimeter as a particular case.
On the one hand, we establish universal $BV$-estimates in every dimension $n \geqslant 2$ for stable sets. Namely, we prove that any stable set in $B_1$ has finite classical perimeter in $B_{1/2}$, with a universal bound. This nonlocal result is new even in the case of $s$-perimeters and its local counterpart (for classical stable minimal surfaces) was known only for simply connected two-dimensional surfaces immersed in $\mathbb{R}^3$.
On the other hand, we prove quantitative flatness estimates for minimizers and stable sets in low dimensions $n = 2, 3$. More precisely, we show that a stable set in $B_R$, with $R$ large, is very close in measure to being a half space in $B_1$ – with a quantitative estimate on the measure of the symmetric difference. As a byproduct, we obtain new classification results for stable sets in the whole plane.
</p>projecteuclid.org/euclid.jdg/1563242471_20190715220128Mon, 15 Jul 2019 22:01 EDTLorentzian Einstein metrics with prescribed conformal infinityhttps://projecteuclid.org/euclid.jdg/1563242472<strong>Alberto Enciso</strong>, <strong>Niky Kamran</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 112, Number 3, 505--554.</p><p><strong>Abstract:</strong><br/>
We prove a local well-posedness theorem for the $(n+1)$-dimensional Einstein equations in Lorentzian signature, with initial data $(\widetilde{g},K)$ whose asymptotic geometry at infinity is similar to that anti-de Sitter (AdS) space, and compatible boundary data $\widehat{g}$ prescribed at the time-like conformal boundary of space-time. More precisely, we consider an $n$-dimensional asymptotically hyperbolic Riemannian manifold $(M, \widetilde{g})$ such that the conformally rescaled metric $x^2 \widetilde{g}$ (with $x$ a boundary defining function) extends to the closure $\overline{M}$ of $M$ as a metric of class $C^{n-1} (\overline{M})$ which is also poly-homogeneous of class $C^p_{\mathrm{polyhom}} (\overline{M})$. Likewise we assume that the conformally rescaled symmetric $(0, 2)$-tensor $x^ 2 K$ extends to $\overline{M}$ as a tensor field of class $C^{n-1} (\overline{M})$ which is polyhomogeneous of class $C^{p-1}_{\mathrm{polyhom}} (\overline{M})$. We assume that the initial data $(\widetilde{g}, K)$ satisfy the Einstein constraint equations and also that the boundary datum is of class $C^p$ on $\partial M \times (-T_0, T_0)$ and satisfies a set of natural compatibility conditions with the initial data. We then prove that there exists an integer $r_n$, depending only on the dimension $n$, such that if $p \geqslant 2q + r_n$, with $q$ a positive integer, then there is $T \gt 0$, depending only on the norms of the initial and boundary data, such that the Einstein equations (1.1) has a unique (up to a diffeomorphism) solution $g$ on $(-T, T) \times M$ with the above initial and boundary data, which is such that $x^2 g \in C^{n-1} ((-T, T) \times \overline{M}) \; \cap \; C^q_{\mathrm{polyhom}} ((-T, T) \times \overline{M})$. Furthermore, if $x^2 \widetilde{g} , x^2 K$ are polyhomogeneous of class $C^{\infty}$ and $\widehat{g}$ is in $C^{\infty} ((-T_0, T_0) \times \partial \overline{M})$, then $x^2 g$ is in $C^{\infty}_{\mathrm{polyhom}} ((-T, T) \times \overline{M})$.
</p>projecteuclid.org/euclid.jdg/1563242472_20190715220128Mon, 15 Jul 2019 22:01 EDTGenus bounds for min-max minimal surfaceshttps://projecteuclid.org/euclid.jdg/1563242473<strong>Daniel Ketover</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 112, Number 3, 555--590.</p><p><strong>Abstract:</strong><br/>
We prove optimal genus bounds for minimal surfaces arising from the min-max construction of Simon–Smith. This confirms a conjecture made by Pitts–Rubinstein in 1986.
</p>projecteuclid.org/euclid.jdg/1563242473_20190715220128Mon, 15 Jul 2019 22:01 EDTNonlinear stability results for the modified Mullins–Sekerka and the surface diffusion flowhttps://projecteuclid.org/euclid.jdg/1567216953<strong>E. Acerbi</strong>, <strong>N. Fusco</strong>, <strong>V. Julin</strong>, <strong>M. Morini</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 113, Number 1, 1--53.</p><p><strong>Abstract:</strong><br/>
It is shown that any three-dimensional periodic configuration that is strictly stable for the area functional is exponentially stable for the surface diffusion flow and for the Mullins–Sekerka or Hele–Shaw flow. The same result holds for three-dimensional periodic configurations that are strictly stable with respect to the sharp-interface Ohta–Kawaski energy. In this case, they are exponentially stable for the so-called modified Mullins–Sekerka flow.
</p>projecteuclid.org/euclid.jdg/1567216953_20190830220250Fri, 30 Aug 2019 22:02 EDTStable blowup for the supercritical Yang–Mills heat flowhttps://projecteuclid.org/euclid.jdg/1567216954<strong>Roland Donninger</strong>, <strong>Birgit Schörkhuber</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 113, Number 1, 55--94.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider the heat flow for Yang–Mills connections on $\mathbb{R}^5 \times SO(5)$. In the $SO(5)$-equivariant setting, the Yang–Mills heat equation reduces to a single semilinear reaction-diffusion equation for which an explicit self-similar blowup solution was found by Weinkove [“Singularity formation in the Yang-Mills flow”, Calc. Var. Partial Differential Equations , 19(2):211–220, 2004]. We prove the nonlinear asymptotic stability of this solution under small perturbations. In particular, we show that there exists an open set of initial conditions in a suitable topology such that the corresponding solutions blow up in finite time and converge to a non-trivial self-similar blowup profile on an unbounded domain. Convergence is obtained in suitable Sobolev norms and in $L^{\infty}$.
</p>projecteuclid.org/euclid.jdg/1567216954_20190830220250Fri, 30 Aug 2019 22:02 EDTMaximizing Steklov eigenvalues on surfaceshttps://projecteuclid.org/euclid.jdg/1567216955<strong>Romain Petrides</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 113, Number 1, 95--188.</p><p><strong>Abstract:</strong><br/>
We study the Steklov eigenvalue functionals $\sigma_k (\Sigma, g) L_g (\partial \Sigma)$ on smooth surfaces with non-empty boundary. We prove that, under some natural gap assumptions, these functionals do admit maximal metrics which come with an associated minimal surface with free boundary from $\Sigma$ into some Euclidean ball, generalizing previous results by Fraser and Schoen in [“Sharp eigenvalue bounds and minimal surfaces in the ball,” Invent. Math. , 203(3):823–890, 2016].
</p>projecteuclid.org/euclid.jdg/1567216955_20190830220250Fri, 30 Aug 2019 22:02 EDTCritical points of the classical Eisenstein series of weight twohttps://projecteuclid.org/euclid.jdg/1571882423<strong>Zhijie Chen</strong>, <strong>Chang-Shou Lin</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 113, Number 2, 189--226.</p><p><strong>Abstract:</strong><br/>
In this paper, we completely determine the critical points of the normalized Eisenstein series $E_{2}(\tau)$ of weight 2. Although $E_{2}(\tau)$ is not a modular form, our result shows that $E_{2}(\tau)$ has at most one critical point in every fundamental domain of the form $\gamma (F_{0})$ of $\Gamma_{0}(2)$, where $\gamma (F_{0})$ are translates of the basic fundamental domain $F_{0}$ via the Möbius transformation of $\gamma \in \Gamma_{0}(2)$. We also give a criteria for such fundamental domain containing a critical point of $E_{2}(\tau)$. Furthermore, under the Möbius transformations of $\Gamma_{0}(2)$ action, all critical points can be mapped into the basic fundamental domain $F_{0}$ and their images in $F_{0}$ give rise to a dense subset of the union of three connected smooth curves in $F_{0}$. A geometric interpretation of these smooth curves is also given. It turns out that these curves coincide with the degeneracy curves of trivial critical points of a multiple Green function related to flat tori.
</p>projecteuclid.org/euclid.jdg/1571882423_20191023220042Wed, 23 Oct 2019 22:00 EDTQuantitative volume space form rigidity under lower Ricci curvature bound Ihttps://projecteuclid.org/euclid.jdg/1571882427<strong>Lina Chen</strong>, <strong>Xiaochun Rongy</strong>, <strong>Shicheng Xuz</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 113, Number 2, 227--272.</p><p><strong>Abstract:</strong><br/>
Let $M$ be a compact $n$-manifold of $\mathrm{Ric}_M \geq (n - 1) H$ ($H$ is a constant). We are concerned with the following space form rigidity: $M$ is isometric to a space form of constant curvature $H$ under either of the following conditions:
(i) There is $ \rho \gt 0$ such that for any $x \in M$, the open $ \rho $-ball at $x^{\ast}$ in the (local) Riemannian universal covering space, $ (U^{\ast}_{\rho} , x^{\ast}) \to (B_{\rho} (x) , x)$, has the maximal volume, i.e., the volume of a $\rho$-ball in the simply connected $n$-space form of curvature $H$.
(ii) For $H = -1$, the volume entropy of $M$ is maximal, i.e., $n - 1$ ([LW1]).
The main results of this paper are quantitative space form rigidity, i.e., statements that $M$ is diffeomorphic and close in the Gromov–Hausdorff topology to a space form of constant curvature $H$, if $M$ almost satisfies, under some additional condition, the above maximal volume condition. For $H = 1$, the quantitative spherical space form rigidity improves and generalizes the diffeomorphic sphere theorem in [CC2].
</p>projecteuclid.org/euclid.jdg/1571882427_20191023220042Wed, 23 Oct 2019 22:00 EDTPlurisubharmonic envelopes and supersolutionshttps://projecteuclid.org/euclid.jdg/1571882428<strong>Vincent Guedj</strong>, <strong>Chinh H. Lu</strong>, <strong>Ahmed Zeriahi</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 113, Number 2, 273--313.</p><p><strong>Abstract:</strong><br/>
We make a systematic study of (quasi-)plurisubharmonic envelopes on compact Kähler manifolds, as well as on domains of $\mathbb{C}^n$, by using and extending an approximation process due to Berman [Ber19]. We show that the quasi-plurisubharmonic envelope of a viscosity super-solution is a pluripotential super-solution of a given complex Monge–Ampère equation. We use these ideas to solve complex Monge–Ampère equations by taking lower envelopes of super-solutions.
</p>projecteuclid.org/euclid.jdg/1571882428_20191023220042Wed, 23 Oct 2019 22:00 EDTConvex $\mathbb{RP}^2$ structures and cubic differentials under neck separationhttps://projecteuclid.org/euclid.jdg/1571882429<strong>John Loftin</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 113, Number 2, 315--383.</p><p><strong>Abstract:</strong><br/>
Let $S$ be a closed oriented surface of genus at least two. Labourie and the author have independently used the theory of hyperbolic affine spheres to find a natural correspondence between convex $\mathbb{RP}^2$ structures on $S$ and pairs $(\Sigma, U)$ consisting of a conformal structure $\Sigma$ on $S$ and a holomorphic cubic differential $U$ over $\Sigma$. We consider geometric limits of convex $\mathbb{RP}^2$ structures on $S$ in which the $\mathbb{RP}^2$ structure degenerates only along a set of simple, non-intersecting, nontrivial, non-homotopic loops $c$. We classify the resulting $\mathbb{RP}^2$ structures on $S - c$ and call them regular convex $\mathbb{RP}^2$ structures. Under a natural topology on the moduli space of all regular convex $\mathbb{RP}^2$ structures on $S$, this space is homeomorphic to the total space of the vector bundle over $\overline{M}_g$ each of whose fibers over a noded Riemann surface is the space of regular cubic differentials. The proof relies on previous techniques of the author, Benoist–Hulin, and Dumas–Wolf, as well as some details due to Wolpert of the geometry of hyperbolic metrics on conformal surfaces in $\overline{M}_g$.
</p>projecteuclid.org/euclid.jdg/1571882429_20191023220042Wed, 23 Oct 2019 22:00 EDTCodimension two holomorphic foliationhttps://projecteuclid.org/euclid.jdg/1573786970<strong>Dominique Cerveau</strong>, <strong>A. Lins Neto</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 113, Number 3, 385--416.</p><p><strong>Abstract:</strong><br/>
This paper is devoted to the study of codimension two holomorphic foliations and distributions. We prove the stability of complete intersection of codimension two distributions and foliations in the local case. Conversely we show the existence of codimension two foliations which are not contained in any codimension one foliation. We study problems related to the singular locus and we classify homogeneous foliations of small degree.
</p>projecteuclid.org/euclid.jdg/1573786970_20191114220315Thu, 14 Nov 2019 22:03 ESTReal Gromov–Witten theory in all genera and real enumerative geometry: computationhttps://projecteuclid.org/euclid.jdg/1573786971<strong>Penka Georgieva</strong>, <strong>Aleksey Zinger</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 113, Number 3, 417--491.</p><p><strong>Abstract:</strong><br/>
Gromov–Witten invariants of real-orientable symplectic manifolds of odd “complex” dimensions; the second part studies the orientations on the moduli spaces of real maps used in constructing these invariants. The present paper applies the results of the latter to obtain quantitative and qualitative conclusions about the invariants defined in the former. After describing large collections of real-orientable symplectic manifolds, we show that the genus $1$ real Gromov–Witten invariants of sufficiently positive almost Kahler threefolds are signed counts of real genus 1 curves only and, thus, provide direct lower bounds for the counts of these curves in such targets. We specify real orientations on the real-orientable complete intersections in projective spaces; the real Gromov–Witten invariants they determine are in a sense canonically determined by the complete intersection itself, (at least) in most cases. We also obtain equivariant localization data that computes the real invariants of projective spaces and determines the contributions from many torus fixed loci for other complete intersections. Our results confirm Walcher’s predictions for the vanishing of these invariants in certain cases and for the localization data in other cases. The localization data is also used to demonstrate the non-triviality of our lower bounds for real curves of genus $1$ in the present paper and of higher genera in a separate paper.
</p>projecteuclid.org/euclid.jdg/1573786971_20191114220315Thu, 14 Nov 2019 22:03 ESTAn integral formula and its applications on sub-static manifoldshttps://projecteuclid.org/euclid.jdg/1573786972<strong>Junfang Li</strong>, <strong>Chao Xia</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 113, Number 3, 493--518.</p><p><strong>Abstract:</strong><br/>
In this article, we first establish the main tool—an integral formula (1.1) for Riemannian manifolds with multiple boundary components (or without boundary). This formula generalizes Reilly’s original formula from [15] and the recent result from [17]. It provides a robust tool for sub-static manifolds regardless of the underlying topology.
Using (1.1) and suitable elliptic PDEs, we prove Heintze–Karcher type inequalities for bounded domains in general sub-static manifolds which recovers some of the results from Brendle [2] as special cases.
On the other hand, we prove a Minkowski inequality for static convex hypersurfaces in a sub-static warped product manifold. Moreover, we obtain an almost Schur lemma for horo-convex hypersurfaces in the hyperbolic space and convex hypersurfaces in the hemi-sphere, which can be viewed as a special Alexandrov–Fenchel inequality.
</p>projecteuclid.org/euclid.jdg/1573786972_20191114220315Thu, 14 Nov 2019 22:03 ESTMinimal hypersurfaces and boundary behavior of compact manifolds with nonnegative scalar curvaturehttps://projecteuclid.org/euclid.jdg/1573786973<strong>Siyuan Lu</strong>, <strong>Pengzi Miao</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 113, Number 3, 519--566.</p><p><strong>Abstract:</strong><br/>
On a compact Riemannian manifold with boundary having positive mean curvature, a fundamental result of Shi and Tam states that, if the manifold has nonnegative scalar curvature and if the boundary is isometric to a strictly convex hypersurface in the Euclidean space, then the total mean curvature of the boundary is no greater than the total mean curvature of the corresponding Euclidean hypersurface. In $3$-dimension, Shi-Tam’s result is known to be equivalent to the Riemannian positive mass theorem.
In this paper, we provide a supplement to Shi–Tam’s result by including the boundary effect of minimal hypersurfaces. More precisely, given a compact manifold $\Omega$ with nonnegative scalar curvature, assuming its boundary consists of two parts, $\Sigma_H$ and $\Sigma_O$, where $\Sigma_H$ is the union of all closed minimal hypersurfaces in $\Omega$ and $\Sigma_O$ is assumed to be isometric to a suitable 2-convex hypersurface $\Sigma$ in a spatial Schwarzschild manifold of mass $m$, we establish an inequality relating $m$, the area of $\Sigma_H$, and two weighted total mean curvatures of $\Sigma_O$ and $\Sigma$.
In $3$-dimension, our inequality has implications to isometric embedding and quasi-local mass problems. In a relativistic context, the result can be interpreted as a quasi-local mass type quantity of $\Sigma_O$ being greater than or equal to the Hawking mass of $\Sigma_H$. We further analyze the limit of this quantity associated with suitably chosen isometric embeddings of large spheres in an asymptotically flat $3$-manifold $M$ into a spatial Schwarzschild manifold. We show that the limit equals the ADM mass of $M$. It follows that our result on the compact manifold $\Omega$ is equivalent to the Riemannian Penrose inequality.
</p>projecteuclid.org/euclid.jdg/1573786973_20191114220315Thu, 14 Nov 2019 22:03 ESTIndex to Volume 113https://projecteuclid.org/euclid.jdg/1573786974<p><strong>Source: </strong>Journal of Differential Geometry, Volume 113, Number 3</p>projecteuclid.org/euclid.jdg/1573786974_20191114220315Thu, 14 Nov 2019 22:03 ESTRational curves on compact Kähler manifoldshttps://projecteuclid.org/euclid.jdg/1577502017<strong>Junyan Cao</strong>, <strong>Andreas Höring</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 114, Number 1, 1--39.</p><p><strong>Abstract:</strong><br/>
Mori’s theorem yields the existence of rational curves on projective manifolds such that the canonical bundle is not nef. In this paper we study compact Kähler manifolds such that the canonical bundle is pseudoeffective, but not nef. We present an inductive argument for the existence of rational curves that uses neither deformation theory nor reduction to positive characteristic. The main tool for this inductive strategy is a weak subadjunction formula for log-canonical centres associated to certain big cohomology classes.
</p>projecteuclid.org/euclid.jdg/1577502017_20191227220046Fri, 27 Dec 2019 22:00 ESTHodge-theoretic mirror symmetry for toric stackshttps://projecteuclid.org/euclid.jdg/1577502022<strong>Tom Coates</strong>, <strong>Alessio Corti</strong>, <strong>Hiroshi Iritani</strong>, <strong>Hsian-Hua Tseng</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 114, Number 1, 41--115.</p><p><strong>Abstract:</strong><br/>
Using the mirror theorem [15], we give a Landau–Ginzburg mirror description for the big equivariant quantum cohomology of toric Deligne–Mumford stacks. More precisely, we prove that the big equivariant quantum $D$-module of a toric Deligne–Mumford stack is isomorphic to the Saito structure associated to the mirror Landau–Ginzburg potential. We give a Gelfand–Kapranov–Zelevinsky (GKZ) style presentation of the quantum $D$-module, and a combinatorial description of quantum cohomology as a quantum Stanley–Reisner ring. We establish the convergence of the mirror isomorphism and of quantum cohomology in the big and equivariant setting.
</p>projecteuclid.org/euclid.jdg/1577502022_20191227220046Fri, 27 Dec 2019 22:00 ESTThe Ricci flow on the sphere with marked pointshttps://projecteuclid.org/euclid.jdg/1577502023<strong>D. H. Phong</strong>, <strong>Jian Song</strong>, <strong>Jacob Sturm</strong>, <strong>Xiaowei Wang</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 114, Number 1, 117--170.</p><p><strong>Abstract:</strong><br/>
The Ricci flow on the $2$-sphere with marked points is shown to converge in all three stable, semi-stable, and unstable cases. In the stable case, the flow was known to converge without any reparametrization, and a new proof of this fact is given. The semistable and unstable cases are new, and it is shown that the flow converges in the Gromov–Hausdorff topology to a limiting metric space which is also a $2$-sphere, but with different marked points and, hence, a different complex structure. The limiting metric is the unique conical constant curvature metric in the semi-stable case, and the unique conical shrinking gradient Ricci soliton metric in the unstable case.
</p>projecteuclid.org/euclid.jdg/1577502023_20191227220046Fri, 27 Dec 2019 22:00 ESTExtremal metrics on blowups along submanifoldshttps://projecteuclid.org/euclid.jdg/1577502024<strong>Reza Seyyedali</strong>, <strong>Gábor Székelyhidi</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 114, Number 1, 171--192.</p><p><strong>Abstract:</strong><br/>
We give conditions under which the blowup of an extremal Kähler manifold along a submanifold of codimension greater than two admits an extremal metric. This generalizes work of Arezzo–Pacard–Singer, who considered blowups in points.
</p>projecteuclid.org/euclid.jdg/1577502024_20191227220046Fri, 27 Dec 2019 22:00 ESTFloer theory for Lagrangian cobordismshttps://projecteuclid.org/euclid.jdg/1583377213<strong>Baptiste Chantraine</strong>, <strong>Georgios Dimitroglou Rizell</strong>, <strong>Paolo Ghiggini</strong>, <strong>Roman Golovko</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 114, Number 3, 393--465.</p><p><strong>Abstract:</strong><br/>
In this article we define intersection Floer homology for exact Lagrangian cobordisms between Legendrian submanifolds in the contactisation of a Liouville manifold, provided that the Chekanov–Eliashberg algebras of the negative ends of the cobordisms admit augmentations. From this theory we derive several long exact sequences relating the Morse homology of an exact Lagrangian cobordism with the bilinearised contact homologies of its ends. These are then used to investigate the topological properties of exact Lagrangian cobordisms.
</p>projecteuclid.org/euclid.jdg/1583377213_20200304220025Wed, 04 Mar 2020 22:00 ESTMinimal surfaces in the $3$-sphere by stacking Clifford torihttps://projecteuclid.org/euclid.jdg/1583377214<strong>David Wiygul</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 114, Number 3, 467--549.</p><p><strong>Abstract:</strong><br/>
Extending work of Kapouleas and Yang, for any integers $N \geq 2, {k , \ell} \geq 1$, and m sufficiently large, we apply gluing methods to construct in the round $3$-sphere a closed embedded minimal surface that has genus ${k \ell m}^2 (N-1)+1$ and is invariant under a $D_{km} \times D_{\ell m}$ subgroup of $O(4)$, where $D_n$ is the dihedral group of order $2n$. Each such surface resembles the union of $N$ nested topological tori, all small perturbations of a single Clifford torus $\mathbb{T}$, that have been connected by ${k \ell m}^2(N-1)$ small catenoidal tunnels, with ${k \ell m}^2$ tunnels joining each pair of neighboring tori. In the large-$m$ limit for fixed $N$, $k$, and $\ell$, the corresponding surfaces converge to $\mathbb{T}$ counted with multiplicity $N$.
</p>projecteuclid.org/euclid.jdg/1583377214_20200304220025Wed, 04 Mar 2020 22:00 ESTOn the entropy of closed hypersurfaces and singular self-shrinkershttps://projecteuclid.org/euclid.jdg/1583377215<strong>Jonathan J. Zhu</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 114, Number 3, 551--593.</p><p><strong>Abstract:</strong><br/>
Self-shrinkers are the special solutions of mean curvature flow in $\mathbf{R}^{n+1}$ that evolve by shrinking homothetically; they serve as singularity models for the flow. The entropy of a hypersurface introduced by Colding–Minicozzi is a Lyapunov functional for the mean curvature flow, and is fundamental to their theory of generic mean curvature flow.
In this paper we prove that a conjecture of Colding–Ilmanen–Minicozzi–White, namely that any closed hypersurface in $\mathbf{R}^{n+1}$ has entropy at least that of the round sphere, holds in any dimension $n$. This result had previously been established for the cases $n \leq 6$ by Bernstein–Wang using a carefully constructed weak flow.
The main technical result of this paper is an extension of Colding–Minicozzi’s classification of entropy-stable self-shrinkers to the singular setting. In particular, we show that any entropystable self-shrinker whose singular set satisfies Wickramasekera’s α-structural hypothesis must be a round cylinder $\mathbf{S}^k (\sqrt{2k}) \times \mathbf{R}^{n-k}$.
</p>projecteuclid.org/euclid.jdg/1583377215_20200304220025Wed, 04 Mar 2020 22:00 ESTIndex to Volume 114https://projecteuclid.org/euclid.jdg/1583377216<p><strong>Source: </strong>Journal of Differential Geometry, Volume 114, Number 3</p>projecteuclid.org/euclid.jdg/1583377216_20200304220025Wed, 04 Mar 2020 22:00 ESTThe catenoid estimate and its geometric applicationshttps://projecteuclid.org/euclid.jdg/1586224840<strong>Daniel Ketover</strong>, <strong>Fernando C. Marques</strong>, <strong>André Neves</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 115, Number 1, 1--26.</p><p><strong>Abstract:</strong><br/>
We prove a sharp area estimate for catenoids that allows us to rule out the phenomenon of multiplicity in min-max theory in several settings. We apply it to prove that i) the width of a three-manifold with positive Ricci curvature is realized by an orientable minimal surface ii) minimal genus Heegaard surfaces in such manifolds can be isotoped to be minimal and iii) the “doublings” of the Clifford torus by Kapouleas–Yang can be constructed variationally by an equivariant min-max procedure. In higher dimensions we also prove that the width of manifolds with positive Ricci curvature is achieved by an index $1$ orientable minimal hypersurface.
</p>projecteuclid.org/euclid.jdg/1586224840_20200406220048Mon, 06 Apr 2020 22:00 EDTAn intrinsic hyperboloid approach for Einstein Klein–Gordon equationshttps://projecteuclid.org/euclid.jdg/1586224841<strong>Qian Wang</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 115, Number 1, 27--109.</p><p><strong>Abstract:</strong><br/>
Klainerman introduced in [7] the hyperboloidal method to prove global existence results for nonlinear Klein–Gordon equations by using commuting vector fields. In this paper, we extend the hyperboloidal method from Minkowski space to Lorentzian spacetimes. This approach is developed in [15] for proving, under the maximal foliation gauge, the global nonlinear stability of Minkowski space for Einstein equations with massive scalar fields, which states that, the sufficiently small data in a compact domain, surrounded by a Schwarzschild metric, leads to a unique, globally hyperbolic, smooth and geodesically complete solution to the Einstein Klein–Gordon system.
In this paper, we set up the geometric framework of the intrinsic hyperboloid approach in the curved spacetime. By performing a thorough geometric comparison between the radial normal vector field induced by the intrinsic hyperboloids and the canonical $\partial_r$, we manage to control the hyperboloids when they are close to their asymptote, which is a light cone in the Schwarzschild zone. By using such geometric information, we not only obtain the crucial boundary information for running the energy method in [15], but also prove that the intrinsic geometric quantities including the Hawking mass all converge to their Schwarzschild values when approaching the asymptote.
</p>projecteuclid.org/euclid.jdg/1586224841_20200406220048Mon, 06 Apr 2020 22:00 EDTThe reverse Yang–Mills–Higgs flow in a neighbourhood of a critical pointhttps://projecteuclid.org/euclid.jdg/1586224842<strong>Graeme Wilkin</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 115, Number 1, 111--174.</p><p><strong>Abstract:</strong><br/>
The main result of this paper is a construction of solutions to the reverse Yang–Mills–Higgs flow converging in the $C^{\infty}$ topology to a critical point. The construction uses only the complex gauge group action, which leads to an algebraic classification of the isomorphism classes of points in the unstable set of a critical point in terms of a filtration of the underlying Higgs bundle.
Analysing the compatibility this filtration with the canonical Harder–Narasimhan–Seshadri double filtration gives an algebraic criterion for two critical points to be connected by a flow line. As an application, we use this to construct Hecke modifications of Higgs bundles via the Yang–Mills–Higgs flow. When the Higgs field is zero (corresponding to the Yang–Mills flow), this criterion has a geometric interpretation in terms of secant varieties of the projectivisation of the underlying bundle inside the unstable manifold of a critical point, which gives a precise description of broken and unbroken flow lines connecting two critical points. For non-zero Higgs field, at generic critical points the analogous interpretation involves the secant varieties of the spectral curve of the Higgs bundle.
</p>projecteuclid.org/euclid.jdg/1586224842_20200406220048Mon, 06 Apr 2020 22:00 EDTOn the global rigidity of sphere packings on $3$-dimensional manifoldshttps://projecteuclid.org/euclid.jdg/1586224843<strong>Xu Xu</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 115, Number 1, 175--193.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove the global rigidity of sphere packings on $3$-dimensional manifolds. This is a $3$-dimensional analogue of the rigidity theorem of Andreev–Thurston and was conjectured by Cooper and Rivin in [5]. We also prove a global rigidity result using a combinatorial scalar curvature introduced by Ge and the author in [13].
</p>projecteuclid.org/euclid.jdg/1586224843_20200406220048Mon, 06 Apr 2020 22:00 EDTThe nonexistence of negative weight derivations on positive dimensional isolated singularities: Generalized Wahl conjecturehttps://projecteuclid.org/euclid.jdg/1589853625<strong>Bingyi Chen</strong>, <strong>Hao Chen</strong>, <strong>Stephen S.-T. Yau</strong>, <strong>Huaiqing Zuo</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 115, Number 2, 195--224.</p><p><strong>Abstract:</strong><br/>
Let $R = \mathbb{C} [ x_1, x_2, \dotsc , x_n ] / (f)$ where $f$ is a weighted homogeneous polynomial defining an isolated singularity at the origin. Then $R$ and $\operatorname{Der} (R,R)$ are graded. It is well-known that $\operatorname{Der} (R,R)$ does not have a negatively graded component. Wahl conjectured that this is still true for $R = \mathbb{C} [ x_1, x_2, \dotsc, x_n] / (f_1, f_2, \dotsc, f_m)$ which defines an isolated, normal and complete intersection singularity and $f_1, f_2, \dotsc, f_m$ weighted homogeneous polynomials with the same weight type $(w_1, w_2, \dotsc, w_n)$. Here we give a positive answer to the Wahl Conjecture and its generalization (without the condition of complete intersection singularity) for $R$ when the degree of $f_i, 1 \leq i \leq m$ are bounded below by a constant $C$ depending only on the weights $w_1, w_2, \dotsc, w_n$. Moreover this bound C is improved when any two of $w_1, w_2, \dotsc, w_n$ are coprime. Since there are counter-examples for the Wahl Conjecture and its generalization when $f_i$ are low degree, our theorem is more or less optimal in the sense that only the lower bound constant can be improved.
</p>projecteuclid.org/euclid.jdg/1589853625_20200518220033Mon, 18 May 2020 22:00 EDTIsoparametric hypersurfaces with four principal curvatures, IVhttps://projecteuclid.org/euclid.jdg/1589853626<strong>Quo-Shin Chi</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 115, Number 2, 225--301.</p><p><strong>Abstract:</strong><br/>
We prove that an isoparametric hypersurface with four principal curvatures and multiplicity pair $(7, 8)$ is either the one constructed by Ozeki and Takeuchi, or one of the two constructed by Ferus, Karcher, and Münzner. This completes the classification of isoparametric hypersurfaces in spheres that É. Cartan initiated in the late 1930s.
</p>projecteuclid.org/euclid.jdg/1589853626_20200518220033Mon, 18 May 2020 22:00 EDTExpanding Kähler–Ricci solitons coming out of Kähler coneshttps://projecteuclid.org/euclid.jdg/1589853627<strong>Ronan J. Conlon</strong>, <strong>Alix Deruelle</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 115, Number 2, 303--365.</p><p><strong>Abstract:</strong><br/>
We give necessary and sufficient conditions for a Kähler equivariant resolution of a Kähler cone, with the resolution satisfying one of a number of auxiliary conditions, to admit a unique asymptotically conical (AC) expanding gradient Kähler–Ricci soliton. In particular, it follows that for any $n \in \mathbb{N}_0$ and for any negative line bundle $L$ over a compact Kähler manifold $D$, the total space of the vector bundle $L^{\oplus (n+1)}$ admits a unique AC expanding gradient Kähler–Ricci soliton with soliton vector field a positive multiple of the Euler vector field if and only if $c_1 \Bigl ( K_D \oplus {(L^\ast)}^{\oplus (n+1)} \Bigr ) \gt 0$. This generalizes the examples already known in the literature. We further prove a general uniqueness result and show that the space of certain AC expanding gradient Kähler–Ricci solitons on $\mathbb{C}^n$ with positive curvature operator on $(1, 1)$-forms is path-connected.
</p>projecteuclid.org/euclid.jdg/1589853627_20200518220033Mon, 18 May 2020 22:00 EDTMinimizing cones associated with isoparametric foliationshttps://projecteuclid.org/euclid.jdg/1589853628<strong>Zizhou Tang</strong>, <strong>Yongsheng Zhang</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 115, Number 2, 367--393.</p><p><strong>Abstract:</strong><br/>
Associated with isoparametric foliations of unit spheres, there are two classes of minimal surfaces − minimal isoparametric hypersurfaces and focal submanifolds. By virtue of their rich structures, we find new series of minimizing cones. They are cones over focal submanifolds and cones over suitable products among these two classes. Except in low dimensions, all such cones are shown minimizing.
</p>projecteuclid.org/euclid.jdg/1589853628_20200518220033Mon, 18 May 2020 22:00 EDTRiemann–Hilbert problems for the resolved conifold and non-perturbative partition functionshttps://projecteuclid.org/euclid.jdg/1594260015<strong>Tom Bridgeland</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 115, Number 3, 395--435.</p><p><strong>Abstract:</strong><br/>
We study the Riemann-Hilbert problems of [6] (T. Bridgeland, “Riemann-Hilbert problems from Donaldson–Thomas theory”, arxiv:1611.03697) in the case of the Donaldson–Thomas theory of the resolved conifold. We give explicit solutions in terms of the Barnes double and triple sine functions. We show that the $\tau$-function of [6] is a non-perturbative partition function, in the sense that its asymptotic expansion coincides with the topological closed string partition function.
</p>projecteuclid.org/euclid.jdg/1594260015_20200708220023Wed, 08 Jul 2020 22:00 EDTVariation of complex structures and variation of Lie algebras II: new Lie algebras arising from singularitieshttps://projecteuclid.org/euclid.jdg/1594260016<strong>Bingyi Chen</strong>, <strong>Naveed Hussain</strong>, <strong>Stephen S.-T. Yau</strong>, <strong>Huaiqing Zuo</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 115, Number 3, 437--473.</p><p><strong>Abstract:</strong><br/>
Finite dimensional Lie algebras are semi-direct product of the semi-simple Lie algebras and solvable Lie algebras. Brieskorn gave the connection between simple Lie algebras and simple singularities. Simple Lie algebras have been well understood, but not the solvable (nilpotent) Lie algebras. It is extremely important to establish connections between singularities and solvable (nilpotent) Lie algebras. In this article, a new natural connection between the set of complex analytic isolated hypersurface singularities and the set of finite dimensional solvable (nilpotent) Lie algebras has been constructed. We construct finite dimensional solvable (nilpotent) Lie algebras naturally from isolated hypersurface singularities. These constructions help us to understand the solvable (nilpotent) Lie algebras from the geometric point of view. Moreover, it is known that the classification of nilpotent Lie algebras in higher dimensions ($\gt 7$) remains to be a vast open area. There are one-parameter families of non-isomorphic nilpotent Lie algebras (but no two-parameter families) in dimension seven. Dimension seven is the watershed of the existence of such families. It is well-known that no such family exists in dimension less than seven, while it is hard to construct one-parameter family in dimension greater than seven. In this article, we construct an one-parameter family of solvable (resp. nilpotent) Lie algebras of dimension $11$ (resp. $10$) from $\tilde{E}_7$ singularities and show that the weak Torelli-type theorem holds. We shall also construct an one-parameter family of solvable (resp. nilpotent) Lie algebras of dimension $12$ (resp. $11$) from $\tilde{E}_8$ singularities and show that the Torelli-type theorem holds. Moreover, we investigate the numerical relation between the dimensions of the new Lie algebras and Yau algebras. Finally, the new Lie algebras arising from fewnomial isolated singularities are also computed.
</p>projecteuclid.org/euclid.jdg/1594260016_20200708220023Wed, 08 Jul 2020 22:00 EDTDeligne pairings and families of rank one local systems on algebraic curveshttps://projecteuclid.org/euclid.jdg/1594260017<strong>Gerard Freixas i Montplet</strong>, <strong>Richard A. Wentworth</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 115, Number 3, 475--528.</p><p><strong>Abstract:</strong><br/>
For smooth families $\mathcal{X} \to S$ of projective algebraic curves and holomorphic line bundles $\mathcal{L, M} \to X$ equipped with flat relative connections, we prove the existence of a canonical and functorial “intersection” connection on the Deligne pairing $\langle \mathcal{L, M} \rangle \to S$. This generalizes the construction of Deligne in the case of Chern connections of hermitian structures on $\mathcal{L}$ and $\mathcal{M}$. A relationship is found with the holomorphic extension of analytic torsion, and in the case of trivial fibrations we show that the Deligne isomorphism is flat with respect to the connections we construct. Finally, we give an application to the construction of a meromorphic connection on the hyperholomorphic line bundle over the twistor space of rank one flat connections on a Riemann surface.
</p>projecteuclid.org/euclid.jdg/1594260017_20200708220023Wed, 08 Jul 2020 22:00 EDTChern–Ricci flows on noncompact complex manifoldshttps://projecteuclid.org/euclid.jdg/1594260018<strong>Man-Chun Lee</strong>, <strong>Luen-Fai Tam</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 115, Number 3, 529--564.</p><p><strong>Abstract:</strong><br/>
In this work, we obtain existence criteria for Chern–Ricci flows on noncompact manifolds. We generalize a result by Tossati–Wienkove [37] on Chern-Ricci flows to noncompact manifolds and a result for Kähler–Ricci flows by Lott–Zhang [21] to Chern–Ricci flows. Using the existence results, we prove that any complete noncollapsed Kähler metric with nonnegative bisectional curvature on a noncompact complex manifold can be deformed to a complete Kähler metric with nonnegative and bounded bisectional curvature which will have maximal volume growth if the initial metric has maximal volume growth. Combining this result with [3], we give another proof that a complete noncompact Kähler manifold with nonnegative bisectional curvature (not necessarily bounded) and maximal volume growth is biholomorphic to $\mathbb{C}^n$. This last result has already been proved by Liu [20] recently using other methods. This last result is a partial confirmation of a uniformization conjecture of Yau [41].
</p>projecteuclid.org/euclid.jdg/1594260018_20200708220023Wed, 08 Jul 2020 22:00 EDTTotally geodesic submanifolds of Teichmüller spacehttps://projecteuclid.org/euclid.jdg/1594260019<strong>Alex Wright</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 115, Number 3, 565--575.</p><p><strong>Abstract:</strong><br/>
Let $\mathcal{T}_{g,n}$ and $\mathcal{M}_{g,n}$ denote the Teichmüller and moduli space respectively of genus $g$ Riemann surfaces with $n$ marked points. The Teichmüller metric on these spaces is a natural Finsler metric that quantifies the failure of two different Riemann surfaces to be conformally equivalent. It is equal to the Kobayashi metric [Roy74], and hence reflects the intrinsic complex geometry of these spaces.
There is a unique holomorphic and isometric embedding from the hyperbolic plane to $\mathcal{T}_{g,n}$ whose image passes through any two given points. The images of such maps, called Teichmüller disks or complex geodesics, are much studied in relation to the geometry and dynamics of Riemann surfaces and their moduli spaces.
A complex submanifold of $\mathcal{T}_{g,n}$ is called totally geodesic if it contains a complex geodesic through any two of its points, and a subvariety of $\mathcal{M}_g$ is called totally geodesic if a component of its preimage in $\mathcal{T}_{g,n}$ is totally geodesic. Totally geodesic submanifolds of dimension $1$ are exactly the complex geodesics.
Almost every complex geodesic in $\mathcal{T}_{g,n}$ has dense image in $\mathcal{M}_{g,n}$ [Mas82, Vee82]. We show that higher dimensional totally geodesic submanifolds are much more rigid.
</p>projecteuclid.org/euclid.jdg/1594260019_20200708220023Wed, 08 Jul 2020 22:00 EDTIndex to Volume 115https://projecteuclid.org/euclid.jdg/1594260020<p><strong>Source: </strong>Journal of Differential Geometry, Volume 115, Number 3</p>projecteuclid.org/euclid.jdg/1594260020_20200708220023Wed, 08 Jul 2020 22:00 EDT