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 EDTResolution of the canonical fiber metrics for a Lefschetz fibrationhttps://projecteuclid.org/euclid.jdg/1518490819<strong>Richard Melrose</strong>, <strong>Xuwen Zhu</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 108, Number 2, 295--317.</p><p><strong>Abstract:</strong><br/>
We consider the family of constant curvature fiber metrics for a Lefschetz fibration with regular fibers of genus greater than one. A result of Obitsu and Wolpert is refined by showing that on an appropriate resolution of the total space, constructed by iterated blow-up, this family is log-smooth, i.e., polyhomogeneous with integral powers but possible multiplicities, at the preimage of the singular fibers in terms of parameters of size comparable to the logarithm of the length of the shrinking geodesic.
</p>projecteuclid.org/euclid.jdg/1518490819_20180212220035Mon, 12 Feb 2018 22:00 ESTThe $C_0$-inextendibility of the Schwarzschild spacetime and the spacelike diameter in Lorentzian geometryhttps://projecteuclid.org/euclid.jdg/1518490820<strong>Jan Sbierski</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 108, Number 2, 319--378.</p><p><strong>Abstract:</strong><br/>
The maximal analytic Schwarzschild spacetime is manifestly inextendible as a Lorentzian manifold with a twice continuously differentiable metric. In this paper, we prove the stronger statement that it is even inextendible as a Lorentzian manifold with a continuous metric. To capture the obstruction to continuous extensions through the curvature singularity, we introduce the notion of the spacelike diameter of a globally hyperbolic region of a Lorentzian manifold with a merely continuous metric and give a sufficient condition for the spacelike diameter to be finite. The investigation of low-regularity inextendibility criteria is motivated by the strong cosmic censorship conjecture.
</p>projecteuclid.org/euclid.jdg/1518490820_20180212220035Mon, 12 Feb 2018 22:00 ESTComparing the Morse index and the first Betti number of minimal hypersurfaceshttps://projecteuclid.org/euclid.jdg/1519959621<strong>Lucas Ambrozio</strong>, <strong>Alessandro Carlotto</strong>, <strong>Ben Sharp</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 108, Number 3, 379--410.</p><p><strong>Abstract:</strong><br/>
By extending and generalizing previous work by Ros and Savo, we describe a method to show that in certain positively curved ambient manifolds the Morse index of every closed minimal hypersurface is bounded from below by a linear function of its first Betti number. The technique is flexible enough to prove that such a relation between the index and the topology of minimal hypersurfaces holds, for example, on all compact rank one symmetric spaces, on products of the circle with spheres of arbitrary dimension and on suitably pinched submanifolds of the Euclidean spaces. These results confirm a general conjecture due to Schoen and Marques–Neves for a wide class of ambient spaces.
</p>projecteuclid.org/euclid.jdg/1519959621_20180301220032Thu, 01 Mar 2018 22:00 ESTOn the Björling problem for Willmore surfaceshttps://projecteuclid.org/euclid.jdg/1519959622<strong>David Brander</strong>, <strong>Peng Wang</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 108, Number 3, 411--457.</p><p><strong>Abstract:</strong><br/>
We solve the analogue of Björling’s problem for Willmore surfaces via a harmonic map representation. For the umbilic-free case the problem and solution are as follows: given a real analytic curve $y_0$ in $\mathbb{S}^3$, together with the prescription of the values of the surface normal and the dual Willmore surface along the curve, lifted to the light cone in Minkowski $5$-space $\mathbb{R}^5_1$, we prove, using isotropic harmonic maps, that there exists a unique pair of dual Willmore surfaces $y$ and $\hat{y}$ satisfying the given values along the curve. We give explicit formulae for the generalized Weierstrass data for the surface pair. For the three dimensional target, we use the solution to explicitly describe the Weierstrass data, in terms of geometric quantities, for all equivariant Willmore surfaces. For the case that the surface has umbilic points, we apply the more general half-isotropic harmonic maps introduced by Hélein to derive a solution: in this case the map $\hat{y}$ is not necessarily the dual surface, and the additional data of a derivative of $\hat{y}$ must be prescribed. This solution is generalized to higher codimensions.
</p>projecteuclid.org/euclid.jdg/1519959622_20180301220032Thu, 01 Mar 2018 22:00 ESTOn the microlocal analysis of the geodesic X-ray transform with conjugate pointshttps://projecteuclid.org/euclid.jdg/1519959623<strong>Sean Holman</strong>, <strong>Gunther Uhlmann</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 108, Number 3, 459--494.</p><p><strong>Abstract:</strong><br/>
We study the microlocal properties of the geodesic X-ray transform $\mathcal{X}$ on a manifold with boundary allowing the presence of conjugate points. Assuming that there are no self-intersecting geodesics and all conjugate pairs are nonsingular we show that the normal operator $\mathcal{N} = \mathcal{X}^t \circ \mathcal{X}$ can be decomposed as the sum of a pseudodifferential operator of order $-1$ and a sum of Fourier integral operators. We also apply this decomposition to prove inversion of $\mathcal{X}$ is only mildly ill-posed when all conjugate points are of order $1$, and a certain graph condition is satisfied, in dimension three or higher.
</p>projecteuclid.org/euclid.jdg/1519959623_20180301220032Thu, 01 Mar 2018 22:00 ESTHeat flows on hyperbolic spaceshttps://projecteuclid.org/euclid.jdg/1519959624<strong>Marius Lemm</strong>, <strong>Vladimir Markovic</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 108, Number 3, 495--529.</p><p><strong>Abstract:</strong><br/>
In this paper we develop new methods for studying the convergence problem for the heat flow on negatively curved spaces and prove that any quasiconformal map of the sphere $\mathbb{S}^{n-1} , n \geq 3$, can be extended to the $n$-dimensional hyperbolic space such that the heat flow starting with this extension converges to a quasi-isometric harmonic map. This implies the Schoen–Li–Wang conjecture that every quasiconformal map of $\mathbb{S}^{n-1} , n \geq 3$, can be extended to a harmonic quasi-isometry of the $n$-dimensional hyperbolic space.
</p>projecteuclid.org/euclid.jdg/1519959624_20180301220032Thu, 01 Mar 2018 22:00 ESTMean curvature flows in manifolds of special holonomyhttps://projecteuclid.org/euclid.jdg/1519959625<strong>Chung-Jun Tsai</strong>, <strong>Mu-Tao Wang</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 108, Number 3, 531--569.</p><p><strong>Abstract:</strong><br/>
We study the uniqueness of minimal submanifolds and the stability of the mean curvature flow in several well-known model spaces of manifolds of special holonomy. These include the Stenzel metric on the cotangent bundle of spheres, the Calabi metric on the cotangent bundle of complex projective spaces, and the Bryant–Salamon metrics on vector bundles over certain Einstein manifolds. In particular, we show that the zero sections, as calibrated submanifolds with respect to their respective ambient metrics, are unique among compact minimal submanifolds and are dynamically stable under the mean curvature flow. The proof relies on intricate interconnections of the Ricci flatness of the ambient space and the extrinsic geometry of the calibrated submanifolds.
</p>projecteuclid.org/euclid.jdg/1519959625_20180301220032Thu, 01 Mar 2018 22:00 ESTIndex to Volume 108https://projecteuclid.org/euclid.jdg/1519959626<p><strong>Source: </strong>Journal of Differential Geometry, Volume 108, Number 3</p>projecteuclid.org/euclid.jdg/1519959626_20180301220032Thu, 01 Mar 2018 22:00 ESTPicard groups of Poisson manifoldshttps://projecteuclid.org/euclid.jdg/1525399215<strong>Henrique Bursztyn</strong>, <strong>Rui Loja Fernandes</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 109, Number 1, 1--38.</p><p><strong>Abstract:</strong><br/>
For a Poisson manifold $M$ we develop systematic methods to compute its Picard group $\mathrm{Pic}(M)$, i.e., its group of self Morita equivalences. We establish a precise relationship between $\mathrm{Pic}(M)$, and the group of gauge transformations up to Poisson diffeomorphisms showing, in particular, that their connected components of the identity coincide; this allows us to introduce the Picard Lie algebra of $M$ and to study its basic properties. Our methods lead to the proof of a conjecture from “Poisson sigma models and symplectic groupoids, in Quantization of Singular Symplectic Quotients” [A.S. Cattaneo, G. Felder, Progress in Mathematics 198 (2001), 41] stating that $\mathrm{Pic}(\mathfrak{g}^*)$, for any compact simple Lie algebra agrees with the group of outer automorphisms of $\mathfrak{g}$.
</p>projecteuclid.org/euclid.jdg/1525399215_20180503220023Thu, 03 May 2018 22:00 EDTRigidity of equality of Lyapunov exponents for geodesic flowshttps://projecteuclid.org/euclid.jdg/1525399216<strong>Clark Butler</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 109, Number 1, 39--79.</p><p><strong>Abstract:</strong><br/>
We study the relationship between the Lyapunov exponents of the geodesic flow of a closed negatively curved manifold and the geometry of the manifold. We show that if each periodic orbit of the geodesic flow has exactly one Lyapunov exponent on the unstable bundle then the manifold has constant negative curvature. We also show under a curvature pinching condition that equality of all Lyapunov exponents with respect to volume on the unstable bundle also implies that the manifold has constant negative curvature. We then study the degree to which one can emulate these rigidity theorems for the hyperbolic spaces of nonconstant negative curvature when the Lyapunov exponents with respect to volume match those of the appropriate symmetric space and obtain rigidity results under additional technical assumptions. The proofs use new results from hyperbolic dynamics including the nonlinear invariance principle of Avila and Viana and the approximation of Lyapunov exponents of invariant measures by Lyapunov exponents associated to periodic orbits which was developed by Kalinin in his proof of the Livsic theorem for matrix cocycles. We also employ rigidity results of Capogna and Pansu on quasiconformal mappings of certain nilpotent Lie groups.
</p>projecteuclid.org/euclid.jdg/1525399216_20180503220023Thu, 03 May 2018 22:00 EDTK-semistability for irregular Sasakian manifoldshttps://projecteuclid.org/euclid.jdg/1525399217<strong>Tristan C. Collins</strong>, <strong>Gábor Székelyhidi</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 109, Number 1, 81--109.</p><p><strong>Abstract:</strong><br/>
We introduce a notion of K-semistability for Sasakian manifolds. This extends to the irregular case of the orbifold K-semistability of Ross–Thomas. Our main result is that a Sasakian manifold with constant scalar curvature is necessarily K-semistable. As an application, we show how one can recover the volume minimization results of Martelli–Sparks–Yau, and the Lichnerowicz obstruction of Gauntlett–Martelli–Sparks–Yau from this point of view.
</p>projecteuclid.org/euclid.jdg/1525399217_20180503220023Thu, 03 May 2018 22:00 EDTNonlocal $s$-minimal surfaces and Lawson coneshttps://projecteuclid.org/euclid.jdg/1525399218<strong>Juan Dávila</strong>, <strong>Manuel del Pino</strong>, <strong>Juncheng Wei</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 109, Number 1, 111--175.</p><p><strong>Abstract:</strong><br/>
The nonlocal $s$-fractional minimal surface equation for $\Sigma = \partial E$ where $E$ is an open set in $\mathbb{R}^N$ is given by
\[ H_\Sigma^ s (p) := \int_{\mathbb{R}^N} \frac{\chi_E(x) - \chi_{E^c}(x)} {{\lvert x-p \rvert}^{N+s}}\, dx \ =\ 0 \; \textrm{for all} \; p\in\Sigma \textrm{ .} \]
Here $0 \lt s \lt 1 , \chi$ designates characteristic function, and the integral is understood in the principal value sense. The classical notion of minimal surface is recovered by letting $s \to 1$. In this paper we exhibit the first concrete examples (beyond the plane) of nonlocal $s$-minimal surfaces. When $s$ is close to $1$, we first construct a connected embedded $s$-minimal surface of revolution in $\mathbb{R}^3$, the nonlocal catenoid, an analog of the standard catenoid $\lvert x_3 \rvert = \log(r+ \sqrt{r^2 - 1})$. Rather than eventual logarithmic growth, this surface becomes asymptotic to the cone $\lvert x_3 \rvert = r \sqrt{1 - s}$. We also find a two-sheet embedded s-minimal surface asymptotic to the same cone, an analog to the simple union of two parallel planes.
On the other hand, for any $0 \lt s \lt 1 , n , m \geq 1$, $s$-minimal Lawson cones $\lvert v \rvert = \alpha \lvert u \rvert, (u, v) \in \mathbb{R}^n \times \mathbb{R}^m$, are found to exist. In sharp contrast with the classical case, we prove their stability for small $s$ and $n + m = 7$, which suggests that unlike the classical theory (or the case $s$ close to $1$), the regularity of $s$-area minimizing surfaces may not hold true in dimension $7$.
</p>projecteuclid.org/euclid.jdg/1525399218_20180503220023Thu, 03 May 2018 22:00 EDTOn realization of tangent cones of homologically area-minimizing compact singular submanifoldshttps://projecteuclid.org/euclid.jdg/1525399219<strong>Yongsheng Zhang</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 109, Number 1, 177--188.</p><p><strong>Abstract:</strong><br/>
We show that every oriented area-minimizing cone in “A Sufficient Criterion for a Cone to be Area-Minimizing” [G.R. Lawlor, Mem. of the Amer. Math. Soc. , Vol. 91, 1991] can be realized as a tangent cone at a singular point of some homologically area-minimizing singular compact submanifold.
</p>projecteuclid.org/euclid.jdg/1525399219_20180503220023Thu, 03 May 2018 22:00 EDTConcerning Toponogov’s theorem and logarithmic improvement of estimates of eigenfunctionshttps://projecteuclid.org/euclid.jdg/1527040871<strong>Matthew D. Blair</strong>, <strong>Christopher D. Sogge</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 109, Number 2, 189--221.</p><p><strong>Abstract:</strong><br/>
We use Toponogov’s triangle comparison theorem from Riemannian geometry along with quantitative scale oriented variants of classical propagation of singularities arguments to obtain logarithmic improvements of the Kakeya–Nikodym norms introduced in [22] for manifolds of nonpositive sectional curvature. Using these and results from our paper [4] we are able to obtain log-improvements of $L^p (M)$ estimates for such manifolds when $2 \lt p \lt \frac{2(n+1)}{n-1}$. These in turn imply $(\log \lambda)^{\sigma_n} , \sigma_n \approx n$, improved lower bounds for $L^1$-norms of eigenfunctions of the estimates of the second author and Zelditch [28], and using a result from Hezari and the second author [18], under this curvature assumption, we are able to improve the lower bounds for the size of nodal sets of Colding and Minicozzi [12] by a factor of $(\log \lambda)^{\mu}$ for any $\mu \lt \frac{2(n+1)^2}{n-1}$, if $n \geq 3$.
</p>projecteuclid.org/euclid.jdg/1527040871_20180522220118Tue, 22 May 2018 22:01 EDTA discrete uniformization theorem for polyhedral surfaceshttps://projecteuclid.org/euclid.jdg/1527040872<strong>Xianfeng David Gu</strong>, <strong>Feng Luo</strong>, <strong>Jian Sun</strong>, <strong>Tianqi Wu</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 109, Number 2, 223--256.</p><p><strong>Abstract:</strong><br/>
A discrete conformality for polyhedral metrics on surfaces is introduced in this paper. It is shown that each polyhedral metric on a compact surface is discrete conformal to a constant curvature polyhedral metric which is unique up to scaling. Furthermore, the constant curvature metric can be found using a finite dimensional variational principle.
</p>projecteuclid.org/euclid.jdg/1527040872_20180522220118Tue, 22 May 2018 22:01 EDTFoliations by spheres with constant expansion for isolated systems without asymptotic symmetryhttps://projecteuclid.org/euclid.jdg/1527040873<strong>Christopher Nerz</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 109, Number 2, 257--289.</p><p><strong>Abstract:</strong><br/>
Motivated by the foliation by stable spheres with constant mean curvature constructed by Huisken–Yau, Metzger proved that every initial data set can be foliated by spheres with constant expansion (CE) if the manifold is asymptotically equal to the standard $[t=0]$-timeslice of the Schwarzschild solution. In this paper, we generalize his result to asymptotically flat initial data sets and weaken additional smallness assumptions made by Metzger. Furthermore, we prove that the CE-surfaces are in a well-defined sense (asymptotically) independent of time if the linear momentum vanishes.
</p>projecteuclid.org/euclid.jdg/1527040873_20180522220118Tue, 22 May 2018 22:01 EDTLG/CY correspondence for elliptic orbifold curves via modularityhttps://projecteuclid.org/euclid.jdg/1527040874<strong>Yefeng Shen</strong>, <strong>Jie Zhou</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 109, Number 2, 291--336.</p><p><strong>Abstract:</strong><br/>
We prove the Landau–Ginzburg/Calabi–Yau correspondence between the Gromov–Witten theory of each elliptic orbifold curve and its Fan–Jarvis–Ruan–Witten theory counterpart via modularity. We show that the correlation functions in these two enumerative theories are different representations of the same set of quasi-modular forms, expanded around different points on the upper-half plane. We relate these two representations by the Cayley transform.
</p>projecteuclid.org/euclid.jdg/1527040874_20180522220118Tue, 22 May 2018 22:01 EDTFully non-linear elliptic equations on compact Hermitian manifoldshttps://projecteuclid.org/euclid.jdg/1527040875<strong>Gábor Székelyhidi</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 109, Number 2, 337--378.</p><p><strong>Abstract:</strong><br/>
We derive a priori estimates for solutions of a general class of fully non-linear equations on compact Hermitian manifolds. Our method is based on ideas that have been used for different specific equations, such as the complex Monge–Ampère, Hessian and inverse Hessian equations. As an application we solve a class of Hessian quotient equations on Kähler manifolds assuming the existence of a suitable subsolution. The method also applies to analogous equations on compact Riemannian manifolds.
</p>projecteuclid.org/euclid.jdg/1527040875_20180522220118Tue, 22 May 2018 22:01 EDTThe geometric torsion conjecture for abelian varieties with real multiplicationhttps://projecteuclid.org/euclid.jdg/1531188186<strong>Benjamin Bakker</strong>, <strong>Jacob Tsimerman</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 109, Number 3, 379--409.</p><p><strong>Abstract:</strong><br/>
The geometric torsion conjecture asserts that the torsion part of the Mordell–Weil group of a family of abelian varieties over a complex quasi-projective curve is uniformly bounded in terms of the genus of the curve. We prove the conjecture for abelian varieties with real multiplication, uniformly in the field of multiplication. Fixing the field, we, furthermore, show that the torsion is bounded in terms of the gonality of the base curve, which is the closer analog of the arithmetic conjecture. The proof is a hybrid technique employing both the hyperbolic and algebraic geometry of the toroidal compactifications of the Hilbert modular varieties $\overline{X}(1)$ parameterizing such abelian varieties. We show that only finitely many torsion covers $\overline{X}_1 (\mathfrak{n})$ contain $d$-gonal curves outside of the boundary for any fixed $d$; the same is true for entire curves $\mathbb{C} \to \overline{X}_1 (\mathfrak{n})$. We also deduce some results about the birational geometry of Hilbert modular varieties.
</p>projecteuclid.org/euclid.jdg/1531188186_20180709220319Mon, 09 Jul 2018 22:03 EDTSubspace concentration of dual curvature measures of symmetric convex bodieshttps://projecteuclid.org/euclid.jdg/1531188189<strong>Károly J. Böröczky</strong>, <strong>Martin Henk</strong>, <strong>Hannes Pollehn</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 109, Number 3, 411--429.</p><p><strong>Abstract:</strong><br/>
We prove a tight subspace concentration inequality for the dual curvature measures of a symmetric convex body.
</p>projecteuclid.org/euclid.jdg/1531188189_20180709220319Mon, 09 Jul 2018 22:03 EDTA discrete uniformization theorem for polyhedral surfaces IIhttps://projecteuclid.org/euclid.jdg/1531188190<strong>Xianfeng Gu</strong>, <strong>Ren Guo</strong>, <strong>Feng Luo</strong>, <strong>Jian Sun</strong>, <strong>Tianqi Wu</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 109, Number 3, 431--466.</p><p><strong>Abstract:</strong><br/>
A notion of discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper. This discrete conformality is shown to be computable. It is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a unique hyperbolic polyhedral metric with a given discrete curvature satisfying Gauss–Bonnet formula. Furthermore, the hyperbolic polyhedral metric with given curvature can be obtained using a discrete Yamabe flow with surgery. In particular, each hyperbolic polyhedral metric on a closed surface with negative Euler characteristic is discrete conformal to a unique hyperbolic metric.
</p>projecteuclid.org/euclid.jdg/1531188190_20180709220319Mon, 09 Jul 2018 22:03 EDTLooijenga’s conjecture via integral-affine geometryhttps://projecteuclid.org/euclid.jdg/1531188193<strong>Philip Engel</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 109, Number 3, 467--495.</p><p><strong>Abstract:</strong><br/>
A cusp singularity is a surface singularity whose minimal resolution is a cycle of smooth rational curves meeting transversely. Cusp singularities come in naturally dual pairs. In 1981, Looijenga proved that whenever a cusp singularity is smoothable, the minimal resolution of the dual cusp is an anticanonical divisor of some smooth rational surface. He conjectured the converse. Recent work of Gross, Hacking, and Keel has proven Looijenga’s conjecture using methods from mirror symmetry. This paper provides an alternative proof of Looijenga’s conjecture based on a combinatorial criterion for smoothability given by Friedman and Miranda in 1983.
</p>projecteuclid.org/euclid.jdg/1531188193_20180709220319Mon, 09 Jul 2018 22:03 EDTThe intersection of a hyperplane with a lightcone in the Minkowski spacetimehttps://projecteuclid.org/euclid.jdg/1531188194<strong>Pengyu Le</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 109, Number 3, 497--507.</p><p><strong>Abstract:</strong><br/>
Klainerman, Luk and Rodnianski derived an anisotropic criterion for formation of trapped surfaces in vacuum, extending the original trapped surface formation theorem of Christodoulou. The effort to understand their result led us to study the intersection of a hyperplane with a lightcone in the Minkowski spacetime. For the intrinsic geometry of the intersection, depending on the hyperplane being spacelike, null or timelike, it has the constant positive, zero or negative Gaussian curvature. For the extrinsic geometry of the intersection, we find that it is a noncompact marginal trapped surface when the hyperplane is null. In this case, we find a geometric interpretation of the Green’s function of the Laplacian on the standard sphere. In the end, we contribute a clearer understanding of the anisotropic criterion for formation of trapped surfaces in vacuum.
</p>projecteuclid.org/euclid.jdg/1531188194_20180709220319Mon, 09 Jul 2018 22:03 EDTThe local picture theorem on the scale of topologyhttps://projecteuclid.org/euclid.jdg/1531188195<strong>William H. Meeks</strong>, <strong>Joaquín Pérez</strong>, <strong>Antonio Ros</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 109, Number 3, 509--565.</p><p><strong>Abstract:</strong><br/>
We prove a descriptive theorem on the extrinsic geometry of an embedded minimal surface of injectivity radius zero in a homogeneously regular Riemannian three-manifold, in a certain small intrinsic neighborhood of a point of almost-minimal injectivity radius . This structure theorem includes a limit object which we call a minimal parking garage structure on $\mathbb{R}^3$, whose theory we also develop.
</p>projecteuclid.org/euclid.jdg/1531188195_20180709220319Mon, 09 Jul 2018 22:03 EDTIndex to Volume 109https://projecteuclid.org/euclid.jdg/1531188196<p><strong>Source: </strong>Journal of Differential Geometry, Volume 109, Number 3</p>projecteuclid.org/euclid.jdg/1531188196_20180709220319Mon, 09 Jul 2018 22:03 EDTThe $L_p$-Aleksandrov problem for $L_p$-integral curvaturehttps://projecteuclid.org/euclid.jdg/1536285625<strong>Yong Huang</strong>, <strong>Erwin Lutwak</strong>, <strong>Deane Yang</strong>, <strong>Gaoyong Zhang</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 1, 1--29.</p><p><strong>Abstract:</strong><br/>
It is shown that within the $L_p$-Brunn–Minkowski theory that Aleksandrov’s integral curvature has a natural $L_p$ extension, for all real $p$. This raises the question of finding necessary and sufficient conditions on a given measure in order for it to be the $L_p$-integral curvature of a convex body. This problem is solved for positive $p$ and is answered for negative $p$ provided the given measure is even.
</p>projecteuclid.org/euclid.jdg/1536285625_20180906220045Thu, 06 Sep 2018 22:00 EDTEntropy of closed surfaces and min-max theoryhttps://projecteuclid.org/euclid.jdg/1536285626<strong>Daniel Ketover</strong>, <strong>Xin Zhou</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 1, 31--71.</p><p><strong>Abstract:</strong><br/>
Entropy is a natural geometric quantity measuring the complexity of a surface embedded in $\mathbb{R}^3$. For dynamical reasons relating to mean curvature flow, Colding–Ilmanen–Minicozzi–White conjectured (since proved by Bernstein–Wang) that the entropy of any closed surface is at least that of the self-shrinking two-sphere. In this paper we give an alternative proof of their conjecture for closed embedded 2-spheres. Our results can be thought of as the parabolic analog to the Willmore conjecture and our argument is analogous in many ways to that of Marques–Neves on the Willmore problem. The main tool is the min-max theory applied to the Gaussian area functional in $\mathbb{R}^3$ which we also establish. To any closed surface in $\mathbb{R}^3$ we associate a four parameter canonical family of surfaces and run a min-max procedure. The key step is ruling out the min-max sequence approaching a self-shrinking plane, and we accomplish this with a degree argument. To establish the min-max theory for $\mathbb{R}^3$ with Gaussian weight, the crucial ingredient is a tightening map that decreases the mass of nonstationary varifolds (with respect to the Gaussian metric of $\mathbb{R}^3$) in a continuous manner.
</p>projecteuclid.org/euclid.jdg/1536285626_20180906220045Thu, 06 Sep 2018 22:00 EDTOn the local extension of the future null infinityhttps://projecteuclid.org/euclid.jdg/1536285627<strong>Junbin Li</strong>, <strong>Xi-Ping Zhu</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 1, 73--133.</p><p><strong>Abstract:</strong><br/>
We consider a characteristic problem of the vacuum Einstein equations with part of the initial data given on a future asymptotically flat null cone, and show that the solution exists uniformly around the null cone for general such initial data. Therefore, the solution contains a piece of the future null infinity. The initial data are not required to be small and the decaying condition is consistent with those in the works of [8] and [11].
</p>projecteuclid.org/euclid.jdg/1536285627_20180906220045Thu, 06 Sep 2018 22:00 EDTWeakly pseudoconvex Kähler manifoldshttps://projecteuclid.org/euclid.jdg/1536285628<strong>Xiangyu Zhou</strong>, <strong>Langfeng Zhu</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 1, 135--186.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove an $L^2$ extension theorem for holomorphic sections of holomorphic line bundles equipped with singular metrics on weakly pseudoconvex Kähler manifolds. Furthermore, in our $L^2$ estimate, optimal constants corresponding to variable denominators are obtained. As applications, we prove an $L^q$ extension theorem with an optimal estimate on weakly pseudoconvex Kähler manifolds and the log-plurisubharmonicity of the fiberwise Bergman kernel in the Kähler case.
</p>projecteuclid.org/euclid.jdg/1536285628_20180906220045Thu, 06 Sep 2018 22:00 EDTThe floating body in real space formshttps://projecteuclid.org/euclid.jdg/1538791243<strong>Florian Besau</strong>, <strong>Elisabeth M. Werner</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 2, 187--220.</p><p><strong>Abstract:</strong><br/>
We carry out a systematic investigation on floating bodies in real space forms. A new unifying approach not only allows us to treat the important classical case of Euclidean space as well as the recent extension to the Euclidean unit sphere, but also the new extension of floating bodies to hyperbolic space.
Our main result establishes a relation between the derivative of the volume of the floating body and a certain surface area measure, which we called the floating area. In the Euclidean setting the floating area coincides with the well known affine surface area, a powerful tool in the affine geometry of convex bodies.
</p>projecteuclid.org/euclid.jdg/1538791243_20181005220130Fri, 05 Oct 2018 22:01 EDTAsymptotics for the wave equation on differential forms on Kerr–de Sitter spacehttps://projecteuclid.org/euclid.jdg/1538791244<strong>Peter Hintz</strong>, <strong>András Vasy</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 2, 221--279.</p><p><strong>Abstract:</strong><br/>
We study asymptotics for solutions of Maxwell’s equations, in fact, of the Hodge–de Rham equation $(d+\delta)u = 0$ without restriction on the form degree, on a geometric class of stationary spacetimes with a warped product type structure (without any symmetry assumptions), which, in particular, include Schwarzschild—de Sitter spaces of all spacetime dimensions $n \geq 4$. We prove that solutions decay exponentially to $0$ or to stationary states in every form degree, and give an interpretation of the stationary states in terms of cohomological information of the spacetime. We also study the wave equation on differential forms and, in particular, prove analogous results on Schwarzschild–de Sitter spacetimes. We demonstrate the stability of our analysis and deduce asymptotics and decay for solutions of Maxwell’s equations, the Hodge–de Rham equation and the wave equation on differential forms on Kerr–de Sitter spacetimes with small angular momentum.
</p>projecteuclid.org/euclid.jdg/1538791244_20181005220130Fri, 05 Oct 2018 22:01 EDTNaturality of Heegaard Floer invariants under positive rational contact surgeryhttps://projecteuclid.org/euclid.jdg/1538791245<strong>Thomas E. Mark</strong>, <strong>Bülent Tosun</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 2, 281--344.</p><p><strong>Abstract:</strong><br/>
For a nullhomologous Legendrian knot in a closed contact $3$-manifold $Y$ we consider a contact structure obtained by positive rational contact surgery. We prove that in this situation the Heegaard Floer contact invariant of $Y$ is mapped by a surgery cobordism to the contact invariant of the result of contact surgery, and we characterize the $\mathrm{spin}^c$ structure on the cobordism that induces the relevant map. As a consequence we determine necessary and sufficient conditions for the nonvanishing of the contact invariant after rational surgery on a Legendrian knot in the standard $3$-sphere, generalizing previous results of Lisca–Stipsicz and Golla. In fact, our methods allow direct calculation of the contact invariant in terms of the rational surgery mapping cone of Ozsváth and Szabó. The proof involves a construction called reducible open book surgery, which reduces in special cases to the capping-off construction studied by Baldwin.
</p>projecteuclid.org/euclid.jdg/1538791245_20181005220130Fri, 05 Oct 2018 22:01 EDTEmbeddedness of least area minimal hypersurfaceshttps://projecteuclid.org/euclid.jdg/1538791246<strong>Antoine Song</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 2, 345--377.</p><p><strong>Abstract:</strong><br/>
In “Simple closed geodesics on convex surfaces” [ J. Differential Geom. , 36(3):517–549, 1992], E. Calabi and J. Cao showed that a closed geodesic of least length in a two-sphere with nonnegative curvature is always simple. Using min-max theory, we prove that for some higher dimensions, this result holds without assumptions on the curvature. More precisely, in a closed $(n+1)$-manifold with $2 \leq n \leq 6$, a least area closed minimal hypersurface exists and any such hypersurface is embedded.
As an application, we give a short proof of the fact that if a closed three-manifold $M$ has scalar curvature at least $6$ and is not isometric to the round three-sphere, then $M$ contains an embedded closed minimal surface of area less than $4 \pi$. This confirms a conjecture of F. C. Marques and A. Neves.
</p>projecteuclid.org/euclid.jdg/1538791246_20181005220130Fri, 05 Oct 2018 22:01 EDTNon-convex balls in the Teichmüller metrichttps://projecteuclid.org/euclid.jdg/1542423625<strong>Maxime Fortier Bourque</strong>, <strong>Kasra Rafi</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 3, 379--412.</p><p><strong>Abstract:</strong><br/>
We prove that the Teichmüller space of surfaces of genus $g$ with $p$ punctures contains balls which are not convex in the Teichmüller metric whenever its complex dimension $(3g −3+p)$ is greater than $1$.
</p>projecteuclid.org/euclid.jdg/1542423625_20181116220051Fri, 16 Nov 2018 22:00 ESTNavigating the space of symmetric CMC surfaceshttps://projecteuclid.org/euclid.jdg/1542423626<strong>Lynn Heller</strong>, <strong>Sebastian Heller</strong>, <strong>Nicholas Schmitt</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 3, 413--455.</p><p><strong>Abstract:</strong><br/>
In this paper we introduce a flow on the spectral data for symmetric CMC surfaces in the $3$-sphere. The flow is designed in such a way that it changes the topology but fixes the intrinsic (metric) and certain extrinsic (periods) closing conditions of the CMC surfaces. By construction the flow yields closed (possibly branched) CMC surfaces at rational times and immersed higher genus CMC surfaces at integer times. We prove the short time existence of this flow near the spectral data of (certain classes of) CMC tori and obtain thereby the existence of new families of closed (possibly branched) connected CMC surfaces of higher genus. Moreover, we prove that flowing the spectral data for the Clifford torus is equivalent to the flow of Plateau solutions by varying the angle of the fundamental piece in Lawson’s construction for the minimal surfaces $\xi_{g,1}$.
</p>projecteuclid.org/euclid.jdg/1542423626_20181116220051Fri, 16 Nov 2018 22:00 ESTInverse problems for the connection Laplacianhttps://projecteuclid.org/euclid.jdg/1542423627<strong>Yaroslav Kurylev</strong>, <strong>Lauri Oksanen</strong>, <strong>Gabriel P. Paternain</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 3, 457--494.</p><p><strong>Abstract:</strong><br/>
We reconstruct a Riemannian manifold and a Hermitian vector bundle with compatible connection from the hyperbolic Dirichlet-to-Neumann operator associated with the wave equation of the connection Laplacian. The boundary data is local and the reconstruction is up to the natural gauge transformations of the problem. As a corollary we derive an elliptic analogue of the main result which solves a Calderón problem for connections on a cylinder.
</p>projecteuclid.org/euclid.jdg/1542423627_20181116220051Fri, 16 Nov 2018 22:00 ESTTowards $A+B$ theory in conifold transitions for Calabi–Yau threefoldshttps://projecteuclid.org/euclid.jdg/1542423628<strong>Yuan-Pin Lee</strong>, <strong>Hui-Wen Lin</strong>, <strong>Chin-Lung Wang</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 3, 495--541.</p><p><strong>Abstract:</strong><br/>
For projective conifold transitions between Calabi–Yau threefolds $X$ and $Y$, with $X$ close to $Y$ in the moduli, we show that the combined information provided by the $A$ model (Gromov–Witten theory in all genera) and $B$ model (variation of Hodge structures) on $X$, linked along the vanishing cycles, determines the corresponding combined information on $Y$. Similar result holds in the reverse direction when linked with the exceptional curves.
</p>projecteuclid.org/euclid.jdg/1542423628_20181116220051Fri, 16 Nov 2018 22:00 ESTExistence of solutions to the even dual Minkowski problemhttps://projecteuclid.org/euclid.jdg/1542423629<strong>Yiming Zhao</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 3, 543--572.</p><p><strong>Abstract:</strong><br/>
Recently, Huang, Lutwak, Yang & Zhang discovered the duals of Federer’s curvature measures within the dual Brunn–Minkowski theory and stated the “Minkowski problem” associated with these new measures. As they showed, this dual Minkowski problem has as special cases the Aleksandrov problem (when the index is $0$) and the logarithmic Minkowski problem (when the index is the dimension of the ambient space)—two problems that were never imagined to be connected in any way. Huang, Lutwak, Yang & Zhang established sufficient conditions to guarantee existence of solution to the dual Minkowski problem in the even setting. In this work, existence of solution to the even dual Minkowski problem is established under new sufficiency conditions. It was recently shown by Böröczky, Henk & Pollehn that these new sufficiency conditions are also necessary.
</p>projecteuclid.org/euclid.jdg/1542423629_20181116220051Fri, 16 Nov 2018 22:00 ESTIndex to Volume 110https://projecteuclid.org/euclid.jdg/1543287636<p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 3</p>projecteuclid.org/euclid.jdg/1543287636_20181126220119Mon, 26 Nov 2018 22:01 ESTEinstein solvmanifolds have maximal symmetryhttps://projecteuclid.org/euclid.jdg/1547607686<strong>Carolyn S. Gordon</strong>, <strong>Michael R. Jablonski</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 1, 1--38.</p><p><strong>Abstract:</strong><br/>
All known examples of homogeneous Einstein metrics of negative Ricci curvature can be realized as left-invariant Riemannian metrics on solvable Lie groups. After defining a notion of maximal symmetry among left-invariant Riemannian metrics on a Lie group, we prove that any left-invariant Einstein metric of negative Ricci curvature on a solvable Lie group is maximally symmetric. This theorem is motivated both by the Alekseevskii Conjecture and by the question of stability of Einstein metrics under the Ricci flow. We also address questions of existence of maximally symmetric left-invariant Riemannian metrics more generally.
</p>projecteuclid.org/euclid.jdg/1547607686_20190115220152Tue, 15 Jan 2019 22:01 ESTOn short time existence for the planar network flowhttps://projecteuclid.org/euclid.jdg/1547607687<strong>Tom Ilmanen</strong>, <strong>André Neves</strong>, <strong>Felix Schulze</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 1, 39--89.</p><p><strong>Abstract:</strong><br/>
We prove the existence of the flow by curvature of regular planar networks starting from an initial network which is non-regular. The proof relies on a monotonicity formula for expanding solutions and a local regularity result for the network flow in the spirit of B. White’s local regularity theorem for mean curvature flow. We also show a pseudolocality theorem for mean curvature flow in any codimension, assuming only that the initial submanifold can be locally written as a graph with sufficiently small Lipschitz constant.
</p>projecteuclid.org/euclid.jdg/1547607687_20190115220152Tue, 15 Jan 2019 22:01 ESTVanishing Pohozaev constant and removability of singularitieshttps://projecteuclid.org/euclid.jdg/1547607688<strong>Jürgen Jost</strong>, <strong>Chunqin Zhou</strong>, <strong>Miaomiao Zhu</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 1, 91--144.</p><p><strong>Abstract:</strong><br/>
Conformal invariance of two-dimensional variational problems is a condition known to enable a blow-up analysis of solutions and to deduce the removability of singularities. In this paper, we identify another condition that is not only sufficient, but also necessary for such a removability of singularities. This is the validity of the Pohozaev identity. In situations where such an identity fails to hold, we introduce a new quantity, called the Pohozaev constant , which on one hand measures the extent to which the Pohozaev identity fails and, on the other hand, provides a characterization of the singular behavior of a solution at an isolated singularity. We apply this to the blow-up analysis for super-Liouville type equations on Riemann surfaces with conical singularities, because in the presence of such singularities, conformal invariance no longer holds and a local singularity is in general non-removable unless the Pohozaev constant is vanishing.
</p>projecteuclid.org/euclid.jdg/1547607688_20190115220152Tue, 15 Jan 2019 22:01 EST$n$-dimension central affine curve flowshttps://projecteuclid.org/euclid.jdg/1547607689<strong>Chuu-Lian Terng</strong>, <strong>Zhiwei Wu</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 1, 145--189.</p><p><strong>Abstract:</strong><br/>
For $n$-dimensional central affine curve flows, we
1) solve the Cauchy problem with periodic initial data and with initial data having rapidly decaying central affine curvatures,
2) construct Bäcklund transformations, a Permutability formula, and explicit solutions,
3) write down formulas for the Bi-Hamiltonian structure and conservation laws.
</p>projecteuclid.org/euclid.jdg/1547607689_20190115220152Tue, 15 Jan 2019 22:01 ESTNull mean curvature flow and outermost MOTShttps://projecteuclid.org/euclid.jdg/1549422101<strong>Theodora Bourni</strong>, <strong>Kristen Moore</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 2, 191--239.</p><p><strong>Abstract:</strong><br/>
We study the evolution of hypersurfaces in spacetime initial data sets by their null mean curvature. A theory of weak solutions is developed using the level-set approach. Starting from an arbitrary mean convex, outer untapped hypersurface $\partial \Omega$, we show that there exists a weak solution to the null mean curvature flow, given as a limit of approximate solutions that are defined using the $\varepsilon$-regularization method. We show that the approximate solutions blow up on the outermost MOTS and the weak solution converges (as boundaries of finite perimeter sets) to a generalized MOTS.
</p>projecteuclid.org/euclid.jdg/1549422101_20190205220216Tue, 05 Feb 2019 22:02 ESTFaltings delta-invariant and semistable degenerationhttps://projecteuclid.org/euclid.jdg/1549422102<strong>Robin de Jong</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 2, 241--301.</p><p><strong>Abstract:</strong><br/>
We determine the asymptotic behavior of the Arakelov metric, the Arakelov–Green’s function, and the Faltings delta-invariant for arbitrary one-parameter families of complex curves with semistable degeneration. The leading terms in the asymptotics are given a combinatorial interpretation in terms of S. Zhang’s theory of admissible Green’s functions on polarized metrized graphs.
</p>projecteuclid.org/euclid.jdg/1549422102_20190205220216Tue, 05 Feb 2019 22:02 ESTQuasi-negative holomorphic sectional curvature and positivity of the canonical bundlehttps://projecteuclid.org/euclid.jdg/1549422103<strong>Simone Diverio</strong>, <strong>Stefano Trapani</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 2, 303--314.</p><p><strong>Abstract:</strong><br/>
We show that if a compact complex manifold admits a Kähler metric whose holomorphic sectional curvature is everywhere non-positive and strictly negative in at least one point, then its canonical bundle is positive. This answers in the affirmative to a question first asked by S.-T. Yau.
</p>projecteuclid.org/euclid.jdg/1549422103_20190205220216Tue, 05 Feb 2019 22:02 ESTOn positive scalar curvature and moduli of curveshttps://projecteuclid.org/euclid.jdg/1549422104<strong>Kefeng Liu</strong>, <strong>Yunhui Wu</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 2, 315--338.</p><p><strong>Abstract:</strong><br/>
In this article we first show that any finite cover of the moduli space of closed Riemann surfaces of genus $g$ with $g \geqslant 2$ does not admit any Riemannian metric $ds^2$ of nonnegative scalar curvature such that ${\lVert \, \cdotp \rVert}_{ds^2} \succ {\lVert \, \cdotp \rVert}_T$ where ${\lVert \, \cdotp \rVert}_T$ is the Teichmüller metric.
Our second result is the proof that any cover $M$ of the moduli space $\mathbb{M}_g$ of a closed Riemann surface $S_g$ does not admit any complete Riemannian metric of uniformly positive scalar curvature in the quasi-isometry class of the Teichmüller metric, which implies a conjecture of Farb–Weinberger in [9].
</p>projecteuclid.org/euclid.jdg/1549422104_20190205220216Tue, 05 Feb 2019 22:02 ESTA 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 EDT