Journal of Differential Geometry Articles (Project Euclid)

Classification of compact ancient solutions to the curve shortening flow

Thu, 05 Aug 2010 15:41 EDT

Panagiota Daskalopoulos, Richard Hamilton, Natasa Sesum

Source: J. Differential Geom., Volume 84, Number 3, 455--464.

Complete Classification of Compact Four-Manifolds with Positive Isotropic Curvature

Tue, 24 Jul 2012 08:41 EDT

Bing-Long Chen, Siu-Hung Tang, Xi-Ping Zhu

Source: J. Differential Geom., Volume 91, Number 1, 41--80.

Abstract:
In this paper, we completely classify all compact 4-manifolds with positive isotropic curvature. We show that they are diffeomorphic to $\mathbb{S}^4$ or $\mathbb{RP}^4$ or quotients of $\mathbb{S}^3 \times \mathbb{R}$ by a cocompact fixed point free subgroup of the isometry group of the standard metric of $\mathbb{S}^3 \times \mathbb{R}$, or a connected sum of them.

On Large Volume Preserving Stable CMC Surfaces in Initial Data Sets

Tue, 24 Jul 2012 08:41 EDT

Michael Eichmair, Jan Metzger

Source: J. Differential Geom., Volume 91, Number 1, 81--102.

Abstract:
Let $(M, g)$ be a complete 3-dimensional asymptotically flat manifold with everywhere positive scalar curvature. We prove that, given a compact subset $K \subset M$, all volume preserving stable constant mean curvature surfaces of sufficiently large area will avoid $K$. This complements the results of G. Huisken and S.-T. Yau and of J. Qing and G. Tian on the uniqueness of large volume preserving stable constant mean curvature spheres in initial data sets that are asymptotically close to Schwarzschild with mass $m \gt 0$. The analysis in G. Huisken and S.-T. Yau and in J. Qing and G. Tian takes place in the asymptotic regime of $M$. Here we adapt ideas from the minimal surface proof of the positive mass theorem by R. Schoen and S.-T. Yau and develop geometric properties of volume preserving stable constant mean curvature surfaces to handle surfaces that run through the part of M that is far from Euclidean.

Semiclassical Spectral Invariants for Schrödinger Operators

Tue, 24 Jul 2012 08:41 EDT

Victor Guillemin, Zuoqin Wang

Source: J. Differential Geom., Volume 91, Number 1, 103--128.

Abstract:
In this article we show how to compute the semiclassical spectral measure associated with the Schrödinger operator on $\mathbb{R}^n$, and, by examining the first few terms in the asymptotic expansion of this measure, obtain inverse spectral results in one and two dimensions. (In particular we show that for the Schrödinger operator on $\mathbb{R}^2$ with a radially symmetric electric potential, $V$, and magnetic potential, $B$, both $V$ and $B$ are spectrally determined.) We also show that in one dimension there is a very simple explicit identity relating the spectral measure of the Schrödinger operator with its Birkhoff canonical form.

The Weil-Petersson Hessian of Length on Teichmüller Space

Tue, 24 Jul 2012 08:41 EDT

Michael Wolf

Source: J. Differential Geom., Volume 91, Number 1, 129--169.

Abstract:
We present a brief but nearly self-contained proof of a formula for the Weil-Petersson Hessian of the geodesic length of a closed curve (either simple or not simple) on a hyperbolic surface. The formula is the sum of the integrals of two naturally defined positive functions over the geodesic, proving convexity of this functional over Teichmüller space (due to Wolpert (1987)). We then estimate this Hessian from below in terms of local quantities and distance along the geodesic. The formula extends to proper arcs on punctured hyperbolic surfaces, and the estimate to laminations. Wolpert’s result that the Thurston metric is a multiple of the Weil-Petersson metric directly follows on taking a limit of the formula over an appropriate sequence of curves. We give further applications to upper bounds of the Hessian, especially near pinching loci, recover through a geometric argument Wolpert’s result on the convexity of length to the half-power, and give a lower bound for growth of length in terms of twist.

Classification of ancient compact solutions to the Ricci flow on surfaces

Wed, 08 Aug 2012 09:00 EDT

Panagiota Daskalopoulos, Richard Hamilton, Natasa Sesum

Source: J. Differential Geom., Volume 91, Number 2, 171--214.

Abstract:
We consider an ancient solution $g(•, t)$ of the Ricci flow on a compact surface that exists for $t\in (−\infty, T)$ and becomes spherical at time $t = T$. We prove that the metric $g(•, t)$ is either a family of contracting spheres, which is a type I ancient solution, or a King–Rosenau solution, which is a type II ancient solution.

Surfaces with parallel mean curvature vector in complex space forms

Wed, 08 Aug 2012 09:00 EDT

Dorel Fetcu

Source: J. Differential Geom., Volume 91, Number 2, 215--232.

Abstract:
We consider surfaces with parallel mean curvature vector (pmc surfaces) in complex space forms and introduce a holomorphic differential on such surfaces. When the complex dimension of the ambient space is equal to two we find a second holomorphic differential and then determine those pmc surfaces on which both differentials vanish. We also provide a reduction of codimension theorem and prove a non-existence result for pmc 2-spheres in complex space forms.

4-manifolds and intersection forms with local coefficients

Wed, 08 Aug 2012 09:00 EDT

Kim A. Frøyshov

Source: J. Differential Geom., Volume 91, Number 2, 233--259.

Abstract:
We extend Donaldson’s diagonalization theorem to intersection forms with certain local coefficients, under some constraints. This provides new examples of non-smoothable topological 4-manifolds.

A problem of Klee on inner section functions of convex bodies

Wed, 08 Aug 2012 09:00 EDT

Richard J. Gardner, Dmitry Ryabogin, Vlad Yaskin, Artem Zvavitch

Source: J. Differential Geom., Volume 91, Number 2, 261--279.

Abstract:
In 1969, Vic Klee asked whether a convex body is uniquely determined (up to translation and reflection in the origin) by its inner section function, the function giving for each direction the maximal area of sections of the body by hyperplanes orthogonal to that direction. We answer this question in the negative by con- structing two infinitely smooth convex bodies of revolution about the $x_n$-axis in $\mathbb{R}^n, n\ge 3$, one origin symmetric and the other not centrally symmetric, with the same inner section function. Moreover, the pair of bodies can be arbitrarily close to the unit ball.

Instantons, concordance, and Whitehead doubling

Wed, 08 Aug 2012 09:00 EDT

Matthew Hedden, Paul Kirk

Source: J. Differential Geom., Volume 91, Number 2, 281--319.

Abstract:
We use moduli spaces of instantons and Chern-Simons invariants of flat connections to prove that the Whitehead doubles of $(2, 2^n − 1)$ torus knots are independent in the smooth knot concordance group; that is, they freely generate a subgroup of infinite rank.

Geodesic-length functions and the Weil-Petersson curvature tensor

Wed, 08 Aug 2012 09:00 EDT

Scott A. Wolpert

Source: J. Differential Geom., Volume 91, Number 2, 321--359.

Abstract:
An expansion is developed for the Weil-Petersson Riemann curvature tensor in the thin region of the Teichmüller and moduli spaces. The tensor is evaluated on the gradients of geodesic lengths for disjoint geodesics. A precise lower bound for sectional curvature in terms of the surface systole is presented. The curvature tensor expansion is applied to establish continuity properties at the frontier strata of the augmented Teichmüller space. The curvature tensor has the asymptotic product structure already observed for the metric and covariant derivative. The product structure is combined with the earlier negative sectional curvature results to establish a classification of asymptotic flats. Furthermore, tangent subspaces of more than half the dimension of Teichmüller space contain sections with a definite amount of negative curvature. Proofs combine estimates for uniformization group exponential-distance sums and potential theory bounds.

Closing Geodesics in $C^1$ Topology

Wed, 03 Oct 2012 15:31 EDT

Ludovic Rifford

Source: J. Differential Geom., Volume 91, Number 3, 361--381.

Abstract:
Given a closed Riemannian manifold, we show how to close an orbit of the geodesic flow by a small perturbation of the metric in the $C^1$ topology.

Cohomology and Hodge Theory on Symplectic Manifolds: I

Wed, 03 Oct 2012 15:31 EDT

Li-Sheng Tseng, Shing-Tung Yau

Source: J. Differential Geom., Volume 91, Number 3, 383--416.

Abstract:
We introduce new finite-dimensional cohomologies on symplectic manifolds. Each exhibits Lefschetz decomposition and contains a unique harmonic representative within each class. Associated with each cohomology is a primitive cohomology defined purely on the space of primitive forms. We identify the dual currents of lagrangians and more generally coisotropic submanifolds with elements of a primitive cohomology, which dualizes to a homology on coisotropic chains.

Cohomology and Hodge Theory on Symplectic Manifolds: II

Wed, 03 Oct 2012 15:31 EDT

Li-Sheng Tseng, Shing-Tung Yau

Source: J. Differential Geom., Volume 91, Number 3, 417--443.

Abstract:
We show that the exterior derivative operator on a symplectic manifold has a natural decomposition into two linear differential operators, analogous to the Dolbeault operators in complex geometry. These operators map primitive forms into primitive forms and therefore lead directly to the construction of primitive cohomologies on symplectic manifolds. Using these operators, we introduce new primitive cohomologies that are analogous to the Dolbeault cohomology in the complex theory. Interestingly, the finiteness of these primitive cohomologies follows directly from an elliptic complex. We calculate the known primitive cohomologies on a nilmanifold and show that their dimensions can vary with the class of the symplectic form.

Yau's Gradient Estimates on Alexandrov Spaces

Wed, 03 Oct 2012 15:31 EDT

Hui-Chun Zhang, Xi-Ping Zhu

Source: J. Differential Geom., Volume 91, Number 3, 445--522.

Abstract:
In this paper, we establish a Bochner-type formula on Alexandrov spaces with Ricci curvature bounded below. Yau’s gradient estimate for harmonic functions is also obtained on Alexandrov spaces.

Critical points of Green's functions on complete manifolds

Tue, 06 Nov 2012 09:13 EST

Alberto Enciso, Daniel Peralta-Salas

Source: J. Differential Geom., Volume 92, Number 1, 1--29.

Abstract:
We prove that the number of critical points of a Li–Tam Green’s function on a complete open Riemannian surface of finite type admits a topological upper bound, given by the first Betti number of the surface. In higher dimensions, we show that there are no topological upper bounds on the number of critical points by constructing, for each nonnegative integer $N$, a Riemannian manifold diffeomorphic to $\mathbb{R}^n\: (n \ge 3)$ whose minimal Green’s function has at least $N$ non-degenerate critical points. Variations on the method of proof of the latter result yield contractible $n$-manifolds whose minimal Green’s functions have level sets diffeomorphic to any fixed codimension 1 compact submanifold of $\mathbb{R}^n$.

Bounding geometry of loops in Alexandrov spaces

Tue, 06 Nov 2012 09:13 EST

Nan Li, Xiaochun Rong

Source: J. Differential Geom., Volume 92, Number 1, 31--54.

Abstract:
For a path in a compact finite dimensional Alexandrov space $X$ with curv $\ge \kappa$, the two basic geometric invariants are the length and the turning angle (which measures the closeness from being a geodesic). We show that the sum of the two invariants of any loop is bounded from below in terms of $\kappa$, the dimension, diameter, and Hausdorff measure of $X$. This generalizes a basic estimate of Cheeger on the length of a closed geodesic in a closed Riemannian manifold. To see that the above result also generalizes and improves an analog of the Cheeger type estimate in Alexandrov geometry in "A.D. Alexandrov spaces with curvature bounded below," we show that for a class of subsets of $X$, the $n$-dimensional Hausdorff measure and rough volume are proportional by a constant depending on $n = \dim(X)$.

Lattice points counting via Einstein metrics

Tue, 06 Nov 2012 09:13 EST

Naichung Conan Leung, Ziming Nikolas Ma

Source: J. Differential Geom., Volume 92, Number 1, 55--69.

Abstract:
We obtain a growth estimate for the number of lattice points inside any $\mathbb{Q}$-Gorenstein cone. Our proof uses the result of Futaki-Ono-Wang on Sasaki-Einstein metric for the toric Sasakian manifold associated to the cone, a Yau’s inequality, and the Kawasaki-Riemann-Roch formula for orbifolds.

Stability of Hodge bundles and a numerical characterization of Shimura varieties

Tue, 06 Nov 2012 09:13 EST

Martin Möller, Eckart Viehweg, Kang Zuo

Source: J. Differential Geom., Volume 92, Number 1, 71--151.

Abstract:
Let $U$ be a connected non-singular quasi-projective variety and $f : A \to U$ a family of abelian varieties of dimension $g$. Suppose that the induced map $U \to \mathcal{A}_g$ is generically finite and there is a compactification $Y$ with complement $S = Y \backslash U$ a normal crossing divisor such that $\Omega_Y^1 (\log S)$ is nef and $\omega_Y (S)$ is ample with respect to $U. We characterize whether $U$ is a Shimura variety by numerical data attached to the variation of Hodge structures, rather than by properties of the map $U \to \mathcal{A}_g$ or by the existence of CM points. More precisely, we show that $f : A \to U$ is a Kuga fibre space, if and only if two conditions hold. First, each irreducible local subsystem $\mathbb{V}$ of $R_1 f_* \mathbb{C}_A$ is either unitary or satisfies the Arakelov equality. Second, for each factor $M$ in the universal cover of $U$ whose tangent bundle behaves like that of a complex ball, an iterated Kodaira-Spencer map associated with $V$ has minimal possible length in the direction of $M$. If in addition $f : A \to U$ is rigid, it is a connected Shimura subvariety of $\mathcal{A}_g$ of Hodge type.

A polynomial bracket for the Dubrovin-Zhang hierarchies

Tue, 06 Nov 2012 09:13 EST

Alexandr Buryak, Hessel Posthuma, Sergey Shadrin

Source: J. Differential Geom., Volume 92, Number 1, 153--185.

Abstract:
We define a hierarchy of Hamiltonian PDEs associated to an arbitrary tau-function in the semi-simple orbit of the Givental group action on genus expansions of Frobenius manifolds. We prove that the equations, the Hamiltonians, and the bracket are weighted-homogeneous polynomials in the derivatives of the dependent variables with respect to the space variable. In the particular case of a conformal (homogeneous) Frobenius structure, our hierarchy coincides with the Dubrovin–Zhang hierarchy that is canonically associated to the underlying Frobenius structure. Therefore, our approach allows to prove the polynomiality of the equations, Hamiltonians, and one of the Poisson brackets of these hierarchies, as conjectured by Dubrovin and Zhang.

On quadratic orthogonal bisectional curvature

Wed, 07 Nov 2012 09:17 EST

Albert Chau, Luen-Fai Tam

Source: J. Differential Geom., Volume 92, Number 1, 187--200.

Abstract:
In this article we study compact Kähler manifolds satisfying a certain nonnegativity condition on the bisectional curvature. Under this condition, we show that the scalar curvature is nonnegative and that the first Chern class is positive assuming local irreducibility. We also obtain a partial classification of possible de Rham decompositions of the universal cover under this condition.

An extension of Schäffer's dual girth conjecture to Grassmannians

Wed, 07 Nov 2012 09:17 EST

Dmitry Faifman

Source: J. Differential Geom., Volume 92, Number 1, 201--220.

Abstract:
In this note we introduce a natural Finsler structure on convex surfaces, referred to as the quotient Finsler structure, which is dual in a sense to the inclusion of a convex surface in a normed space as a submanifold. It has an associated quotient girth, which is similar to the notion of girth defined by Schäffer. We prove the analogs of Schäffer’s dual girth conjecture (proved by Álvarez-Paiva) and the Holmes–Thompson dual volumes theorem in the quotient setting. We then show that the quotient Finsler structure admits a natural extension to higher Grassmannians, and prove the corresponding theorems in the general case. We follow Álvarez-Paiva’s approach to the problem, namely, we study the symplectic geometry of the associated co-ball bundles. For the higher Grassmannians, the theory of Hamiltonian actions is applied.

Proof of the Yano-Obata conjecture for $h$-projective transformations

Wed, 07 Nov 2012 09:17 EST

Vladimir S. Matveev, Stefan Rosemann

Source: J. Differential Geom., Volume 92, Number 1, 221--261.

Abstract:
We prove the classical Yano-Obata conjecture by showing that the connected component of the group of $h$-projective transformations of a closed, connected Riemannian Kähler manifold consists of isometries unless the manifold is the complex projective space with the standard Fubini-Study metric (up to a constant).

Volume inequalities for asymmetric Wulff shapes

Wed, 07 Nov 2012 09:17 EST

Franz E. Schuster, Manuel Weberndorfer

Source: J. Differential Geom., Volume 92, Number 1, 263--283.

Abstract:
Sharp reverse affine isoperimetric inequalities for asymmetric Wulff shapes and their polars are established, along with the characterization of all extremals. These new inequalities have as special cases previously obtained simplex inequalities by Ball, Barthe and Lutwak, Yang, and Zhang. In particular, they provide the solution to a problem by Zhang.

Stable pairs on local $K3$ surfaces

Wed, 07 Nov 2012 09:17 EST

Yukinobu Toda

Source: J. Differential Geom., Volume 92, Number 1, 285--370.

Abstract:
We prove a formula which relates Euler characteristic of moduli spaces of stable pairs on local $K3$ surfaces to counting invariants of semistable sheaves on them. Our formula generalizes Kawai- Yoshioka’s formula for stable pairs with irreducible curve classes to arbitrary curve classes. We also propose a conjectural multiple cover formula of sheaf counting invariants which, combined with our main result, leads to an Euler characteristic version of Katz- Klemm-Vafa conjecture for stable pairs.

Small-time heat kernel asymptotics at the sub-Riemannian cut locus

Wed, 28 Nov 2012 08:43 EST

Davide Barilari, Ugo Boscain, Robert W. Neel

Source: J. Differential Geom., Volume 92, Number 3, 373--416.

Abstract:
For a sub-Riemannian manifold provided with a smooth volume, we relate the small-time asymptotics of the heat kernel at a point y of the cut locus from $x$ with roughly "how much" $y$ is conjugate to $x$. This is done under the hypothesis that all minimizers connecting $x$ to $y$ are strongly normal, i.e. all pieces of the trajectory are not abnormal. Our result is a refinement of the one of Leandre $4t \log p_t(x, y) \to −d^2(x, y)$ for $t \to 0$, in which only the leading exponential term is detected. Our results are obtained by extending an idea of Molchanov from the Riemannian to the sub-Riemannian case, and some details we get appear to be new even in the Riemannian context. These results permit us to obtain properties of the sub-Riemannian distance starting from those of the heat kernel and vice versa. For the Grushin plane endowed with the Euclidean volume, we get the expansion $p_t(x, y) \sim t^{−5/4} \exp(−d^2(x, y)/4t)$ where $y$ is reached from a Riemannian point $x$ by a minimizing geodesic which is conjugate at $y$.

A normal form theorem around symplectic leaves

Wed, 28 Nov 2012 08:43 EST

Marius Crainic, Ioan Mǎrcuţ

Source: J. Differential Geom., Volume 92, Number 3, 417--461.

Abstract:
We prove the Poisson geometric version of the Local Reeb Stability (from foliation theory) and of the Slice Theorem (from equivariant geometry), which is also a generalization of Conn’s linearization theorem.

Parallel tractor extension and ambient metrics of holonomy split $G_2$

Wed, 28 Nov 2012 08:43 EST

C. Robin Graham, Travis Willse

Source: J. Differential Geom., Volume 92, Number 3, 463--506.

Abstract:
The holonomy of the ambient metrics of Nurowski’s conformal structures associated to generic real-analytic 2-plane fields on oriented 5-manifolds is investigated. It is shown that the holonomy is always contained in the split real form $G_2$ of the exceptional Lie group, and is equal to $G_2$ for an open dense set of 2-plane fields given by explicit conditions. In particular, this gives an infinite-dimensional family of metrics of holonomy equal to split $G_2$. These results generalize work of Leistner-Nurowski. The inclusion of the holonomy in $G_2$ is established by proving an ambient extension theorem for parallel tractors in the context of conformal geometry in general signature and dimension, which is expected to be of independent interest.

The sphere theorems for manifolds with positive scalar curvature

Wed, 28 Nov 2012 08:43 EST

Juan-Ru Gu, Hong-Wei Xu

Source: J. Differential Geom., Volume 92, Number 3, 507--545.

Abstract:
Some new differentiable sphere theorems are obtained via the Ricci flow and stable currents. We prove that if $M^n$ is a compact manifold whose normalized scalar curvature and sectional curvature satisfy the pointwise pinching condition $R_0 \gt \sigma_n K_{\rm max}$, where $\sigma_n \in (\frac{1}{4} , 1)$ is an explicit positive constant, then $M$ is diffeomorphic to a spherical space form. We also provide a partial answer to Yau’s conjecture on the pinching theorem. Moreover, we prove that if $M^n(n \ge 3)$ is a compact manifold whose $(n − 2)$-th Ricci curvature and normalized scalar curvature satisfy the pointwise condition $Ric^{(n−2)}_{\rm min} \gt \tau_n(n −2)R_0$, where $\tau_n \in(\frac{1}{4} , 1)$ is an explicit positive constant, then $M$ is diffeomorphic to a spherical space form. We then extend the sphere theorems above to submanifolds in a Riemannian manifold. Finally we give a classification of submanifolds with weakly pinched curvatures, which improves the differentiable pinching theorems due to Andrews, Baker, and the authors.

Cork twisting exotic Stein 4-manifolds

Wed, 02 Jan 2013 10:45 EST

Selman Akbulut, Kouichi Yasui

Source: J. Differential Geom., Volume 93, Number 1, 1--36.

Abstract:
From any 4-dimensional oriented handlebody $X$ without 3- and 4-handles and with $b_2 \ge 1$, we construct arbitrary many compact Stein 4-manifolds that are mutually homeomorphic but not diffeomorphic to each other, so that their topological invariants (their fundamental groups, homology groups, boundary homology groups, and intersection forms) coincide with those of $X$. We also discuss the induced contact structures on their boundaries. Furthermore, for any smooth 4-manifold pair $(Z, Y)$ such that the complement $Z − \operatorname{int} Y$ is a handlebody without 3- and 4-handles and with $b_2 \ge 1$, we construct arbitrary many exotic embeddings of a compact 4-manifold $Y'$ into $Z$, such that $Y'$ has the same topological invariants as $Y$.

The geometric Cauchy problem for surfaces with Lorentzian harmonic Gauss maps

Wed, 02 Jan 2013 10:45 EST

David Brander, Martin Svensson

Source: J. Differential Geom., Volume 93, Number 1, 37--66.

Abstract:
The geometric Cauchy problem for a class of surfaces in a pseudo-Riemannian manifold of dimension 3 is to find the surface which contains a given curve with a prescribed tangent bundle along the curve. We consider this problem for constant negative Gauss curvature surfaces (pseudospherical surfaces) in Euclidean 3-space, and for timelike constant non-zero mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space. We prove that there is a unique solution if the prescribed curve is non-characteristic, and for characteristic initial curves (asymptotic curves for pseudospherical surfaces and null curves for timelike CMC) it is necessary and suffcient for similar data to be prescribed along an additional characteristic curve that intersects the first. The proofs also give a means of constructing all solutions using loop group techniques. The method used is the infinite dimensional d'Alembert type representation for surfaces associated with Lorentzian harmonic maps (1-1 wave maps) into symmetric spaces, developed since the 1990's. Explicit formulae for the potentials in terms of the prescribed data are given, and some applications are considered.

Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces of finite total curvature in $\mathbb{R}^3$

Wed, 02 Jan 2013 10:45 EST

Manuel del Pino, Michal Kowalczyk, Juncheng Wei

Source: J. Differential Geom., Volume 93, Number 1, 67--131.

Abstract:
We consider minimal surfaces $M$ which are complete, embedded, and have finite total curvature in $\mathbb{R}^3$, and bounded, entire solutions with finite Morse index of the Allen-Cahn equation $\Delta u+ f(u) = 0$ in $\mathbb{R}^3$. Here $f = −W'$ with $W$ bi-stable and balanced, for instance $W(u) = \frac{1}{4} (1 − u^2)^2$. We assume that $M$ has $m \ge 2$ ends, and additionally that $M$ is non-degenerate, in the sense that its bounded Jacobi fields are all originated from rigid motions (this is known for instance for a Catenoid and for the Costa-Hoffman-Meeks surface of any genus). We prove that for any small $\alpha \gt 0$, the Allen-Cahn equation has a family of bounded solutions depending on $m − 1$ parameters distinct from rigid motions, whose level sets are embedded surfaces lying close to the blown-up surface $M_\alpha := \alpha^{−1}M$, with ends possibly diverging logarithmically from $M_\alpha$. We prove that these solutions are $L^\infty$- non-degenerate up to rigid motions, and find that their Morse index coincides with the index of the minimal surface. Our construction suggests parallels of De Giorgi conjecture for general bounded solutions of finite Morse index.

Resonance for loop homology of spheres

Wed, 02 Jan 2013 10:45 EST

Nancy Hingston, Hans-Bert Rademacher

Source: J. Differential Geom., Volume 93, Number 1, 133--174.

Abstract:
A Riemannian or Finsler metric on a compact manifold $M$ gives rise to a length function on the free loop space $\Lambda M$, whose critical points are the closed geodesics in the given metric. If $X$ is a homology class on $\Lambda M$, the “minimax” critical level $\mathsf{cr}(X)$ is a critical value. Let $M$ be a sphere of dimension $\gt 2$, and fix a metric $g$ and a coefficient field $G$. We prove that the limit as $\deg(X)$ goes to infinity of $\mathsf{cr}(X)/ \deg(X)$ exists. We call this limit $\overline\alpha = \overline\alpha(M, g,G)$ the global mean frequency of $M$. As a consequence we derive resonance statements for closed geodesics on spheres; in particular either all homology on $\Lambda$ of sufficiently high degreee lies hanging on closed geodesics of mean frequency (length/average index) $\overline\alpha$, or there is a sequence of infinitely many closed geodesics whose mean frequencies converge to $\overline\alpha$. The proof uses the Chas-Sullivan product and results of Goresky-Hingston.

Volume estimates for Kähler-Einstein metrics: The three-dimensional case

Mon, 25 Feb 2013 09:01 EST

X.-X. Chen, S. K. Donaldson

Source: J. Differential Geom., Volume 93, Number 1, 175--189.

Abstract:
We obtain an estimate for the volumes of neighborhoods of sets of large curvature in three-dimensional Kähler-Einstein manifolds. The key technical step is to prove that a version of monotonicity for $L^2$ energy holds as long as the underlying region does not “carry homology” (in the sense that the normalized energy in a ball controls the normalized energy in an interior ball).

Volume estimates for Kähler-Einstein metrics and rigidity of complex structures

Mon, 25 Feb 2013 09:01 EST

X.-X. Chen, S. K. Donaldson

Source: J. Differential Geom., Volume 93, Number 1, 191--201.

Abstract:
This paper extends our earlier results to higher dimensions using a different approach, based on the rigidity of complex structures on certain domains. We prove a “low energy” result in all dimensions, in the sense that if normalized energy in a large ball is small enough, then the normalized energy in any interior ball must also be small.

The completion of the manifold of Riemannian metrics

Mon, 25 Feb 2013 09:01 EST

Brian Clarke

Source: J. Differential Geom., Volume 93, Number 1, 203--268.

Abstract:
We give a description of the completion of the manifold of all smooth Riemannian metrics on a fixed smooth, closed, finitedimensional, orientable manifold with respect to a natural metric called the $L^2$ metric. The primary motivation for studying this problem comes from Teichmüller theory, where similar considerations lead to a completion of the well-known Weil-Petersson metric. We give an application of the main theorem to the completions of Teichmüller space with respect to a class of metrics that generalize the Weil-Petersson metric.

Minimal models, formality, and hard Lefschetz properties of solvmanifolds with local systems

Mon, 25 Feb 2013 09:01 EST

Hisashi Kasuya

Source: J. Differential Geom., Volume 93, Number 1, 269--297.

Abstract:
For a simply connected solvable Lie group $G$ with a cocompact discrete subgroup $\Gamma$, we consider the space of differential forms on the solvmanifold $G/\Gamma$ with values in a certain flat bundle so that this space has a structure of a differential graded algebra (DGA). We construct Sullivan’s minimal model of this DGA. This result is an extension of Nomizu’s theorem for ordinary coefficients in the nilpotent case. By using this result, we refine Hasegawa’s result of formality of nilmanifolds and Benson-Gordon’s result of hard Lefschetz properties of nilmanifolds.

Volume optimization, normal surfaces, and Thurston's equation on triangulated 3-manifolds

Mon, 25 Feb 2013 09:01 EST

Feng Luo

Source: J. Differential Geom., Volume 93, Number 1, 299--326.

Abstract:
We propose a finite-dimensional variational principle on triangulated 3-manifolds so that its critical points are related to solutions to Thurston’s gluing equation and Haken’s normal surface equation. The action functional is the volume.

The Smale conjecture for Seifert fibered spaces with hyperbolic base orbifold

Mon, 25 Feb 2013 09:01 EST

Darryl McCullough, Teruhiko Soma

Source: J. Differential Geom., Volume 93, Number 1, 327--353.

Abstract:
Let $M$ be a closed orientable 3-manifold admitting an $\mathbb{H}^2 \times \mathbb{R}$ or $\widetilde{\mathrm{SL}_2}(\mathbb{R})$ geometry, or equivalently a Seifert fibered space with a hyperbolic base 2-orbifold. Our main result is that the connected component of the identity map in the diffeomorphism group $\mathrm{Diff}(M)$ is either contractible or homotopy equivalent to $S^1$, according as the center of $\pi_1(M)$ is trivial or infinite cyclic. Apart from the remaining case of non-Haken infranilmanifolds, this completes the homeomorphism classifications of $\mathrm{Diff}(M)$ and of the space of Seifert fiberings $\mathrm{SF}(M)$ for compact orientable aspherical 3-manifolds. We also prove that when $M$ has an $\mathbb{H}^2 \times \mathbb{R}$ or $\widetilde{\mathrm{SL}_2}(\mathbb{R})$ geometry and the base orbifold has underlying manifold the 2-sphere with three cone points, the inclusion $\mathrm{Isom}(M) \to \mathrm{Diff}(M)$ is a homotopy equivalence.

Algebraic hyperbolicity of ramified covers of $\mathbb{G}^2_m$ (and integrap points on affine subsets of $\mathbb{P}_2$)

Mon, 25 Feb 2013 21:15 EST

Pietro Corvaja, Umberto Zannier

Source: J. Differential Geom., Volume 93, Number 3, 355--377.

Abstract:
Let $X$ be a smooth affine surface, $X \to \mathbb{G}^2_m$ be a finite morphism. We study the affine curves on $X$, with bounded genus and number of points at infinity, obtaining bounds for their degree in terms of Euler characteristic. A typical example where these bounds hold is represented by the complement of a three-component curve in the projective plane, of total degree at least 4. The corresponding results may be interpreted as bounding the height of integral points on $X$ over a function field. In the language of Diophantine Equations, our results may be rephrased in terms of bounding the height of the solutions of $f(u, v, y) = 0$, with $u, v, y$ over a function field, $u, v$ $S$-units. It turns out that all of this contain some cases of a strong version of a conjecture of Vojta over function fields in the split case. Moreover, our method would apply also to the nonsplit case. We remark that special cases of our results in the holomorphic context were studied by M. Green already in the seventies, and recently in greater generality by Noguchi, Winkelmann, and Yamanoi; however, the algebraic context was left open and seems not to fall in the existing techniques.

On a generalization of a theorem of McDuff

Mon, 25 Feb 2013 21:15 EST

G. Deltour

Source: J. Differential Geom., Volume 93, Number 3, 379--400.

Abstract:
We study the symplectic structure of the holomorphic coadjoint orbits, generalizing a theorem of McDuff on the symplectic structure of Hermitian symmetric spaces of noncompact type.

Quasigeodesic flows and Möbius-like groups

Mon, 25 Feb 2013 21:15 EST

Steven Frankel

Source: J. Differential Geom., Volume 93, Number 3, 401--429.

Abstract:
If M is a hyperbolic 3-manifold with a quasigeodesic flow, then we show that $\pi_1(M)$ acts in a natural way on a closed disc by homeomorphisms. Consequently, such a flow either has a closed orbit or the action on the boundary circle is Möbius-like but not conjugate into $PSL(2,\mathbb{R})$.We conjecture that the latter possibility cannot occur.

Bernstein theorem and regularity for a class of Monge-Ampère equations

Mon, 25 Feb 2013 21:15 EST

Huaiyu Jian, Xu-Jia Wang

Source: J. Differential Geom., Volume 93, Number 3, 431--469.

Abstract:
In this paper we first introduce a transform for convex functions and use it to prove a Bernstein theorem for a Monge-Ampère equation in half space.We then prove the optimal global regularity for a class of Monge-Ampère type equations arising in a number of geometric problems such as Poincaré metrics, hyperbolic affine spheres, and Minkowski type problems.

Minimizers of the Willmore functional under fixed conformal class

Mon, 25 Feb 2013 21:15 EST

Ernst Kuwert , Reiner Schätzle

Source: J. Differential Geom., Volume 93, Number 3, 471--530.

Abstract:
We prove the existence of a smooth minimizer of the Willmore energy in the class of conformal immersions of a given closed Riemann surface into $\mathbb{R}^n, n = 3, 4$, if there is one conformal immersion with Willmore energy smaller than a certain bound $\mathcal{W}_{n,p}$ depending on codimension and genus $p$ of the Riemann surface. For tori in codimension 1, we know $\mathcal{W}_{3,1} = 8\pi$.

Smooth Yamabe invariant and surgery

Tue, 26 Feb 2013 09:31 EST

Bernd Ammann, Mattias Dahl, Emmanuel Humbert

Source: J. Differential Geom., Volume 94, Number 1, 1--58.

Abstract:
We prove a surgery formula for the smooth Yamabe invariant $\sigma(M)$ of a compact manifold $M$. Assume that $N$ is obtained from $M$ by surgery of codimension at least 3. We prove the existence of a positive constant $\Lambda_n$, depending only on the dimension n of $M$, such that \[ \sigma(N) \ge \mathrm{min}\{\sigma(M),\Lambda_n\}.\]

On the dimension datum of a subgroup and its application to isospectral manifolds

Tue, 26 Feb 2013 09:31 EST

Jinpeng An, Jiu-Kang Yu, Jun Yu

Source: J. Differential Geom., Volume 94, Number 1, 59--85.

Abstract:
The dimension datum of a subgroup of a compact Lie group is a piece of spectral information about that subgroup. We find some new invariants and phenomena of the dimension data and apply them to construct the first example of a pair of isospectral, simply connected closed Riemannian manifolds which are of different homotopy types. We also answer questions proposed by Langlands.

Linking, twisting, writing, and helicity on the 2-sphere and in hyperbolic 3-space

Tue, 26 Feb 2013 09:31 EST

Dennis DeTurck, Herman Gluck

Source: J. Differential Geom., Volume 94, Number 1, 87--128.

Abstract:
In the first paper of this series, “Electrodynamics and the Gauss Linking Integral on the 3-sphere and in Hyperbolic 3-space,” we developed a steady-state version of classical electrodynamics in these two spaces, including explicit formulas for the vector-valued Green’s operator, explicit formulas of Biot-Savart type for the magnetic field, and a corresponding Ampère’s Law contained in Maxwell’s equations, and then used these to obtain explicit integral formulas for the linking number of two disjoint closed curves. In this second paper, we obtain integral formulas for twisting, writhing, and helicity, and prove that link = twist + writhe on the 3-sphere and in hyperbolic 3-space. We then use these results to derive upper bounds for the helicity of vector fields and lower bounds for the first eigenvalue of the curl operator on subdomains of these two spaces. An announcement of these results, and a hint of their proofs, can be found in the Math ArXiv, math.GT/0406276, while an expanded version of the first paper, with full proofs, can be found at math.GT/0510388.

Zero-energy fields on complex projective space

Tue, 26 Feb 2013 09:31 EST

Michael Eastwood, Hubert Goldschmidt

Source: J. Differential Geom., Volume 94, Number 1, 129--157.

Abstract:
We consider complex projective space with its Fubini–Study metric and the X-ray transform defined by integration over its geodesics. We identify the kernel of this transform acting on symmetric tensor fields.

Large isopserimetric surfaces in initial data sets

Tue, 26 Feb 2013 09:31 EST

Michael Eichmair, Jan Metzger

Source: J. Differential Geom., Volume 94, Number 1, 159--186.

Abstract:
We study the isoperimetric structure of asymptotically flat Riemannian 3-manifolds $(M, g)$ that are $\mathcal{C}^0$-asymptotic to Schwarzschild of mass $m \gt 0$. Refining an argument due to H. Bray, we obtain an effective volume comparison theorem in Schwarzschild. We use it to show that isoperimetric regions exist in $(M, g)$ for all sufficiently large volumes, and that they are close to centered coordinate spheres. This implies that the volume-preserving stable constant mean curvature spheres constructed by G. Huisken and S.-T. Yau as well as R. Ye as perturbations of large centered coordinate spheres minimize area among all competing surfaces that enclose the same volume. This confirms a conjecture of H. Bray. Our results are consistent with the uniqueness results for volume-preserving stable constant mean curvature surfaces in initial data sets obtained by G. Huisken and S.-T. Yau and strengthened by J. Qing and G. Tian. The additional hypotheses that the surfaces be spherical and far out in the asymptotic region in their results are not necessary in our work.

Sobolev Metrics on the Manifold of All Reimannian Metrics

Wed, 01 May 2013 16:04 EDT

Martin Bauer, Philipp Harms, Peter W. Michor

Source: J. Differential Geom., Volume 94, Number 2, 187--208.

Abstract:
On the manifold $\mathcal{M}(M)$ of all Riemannian metrics on a compact manifold $M$, one can consider the natural $L^2$-metric as described first by D.G. Ebin, The manifold of Riemannian metrics . In this paper we consider variants of this metric, which in general are of higher order. We derive the geodesic equations; we show that they are well-posed under some conditions and induce a locally diffeomorphic geodesic exponential mapping. We give a condition when Ricci flow is a gradient flow for one of these metrics.

Poisson 2-Groups

Wed, 01 May 2013 16:04 EDT

Zhuo Chen, Mathieu Stiçnon, Ping Xu

Source: J. Differential Geom., Volume 94, Number 2, 209--240.

Abstract:
We prove a 2-categorical analogue of a classical result of Drinfeld: there is a one-to-one correspondence between connected, simply connected Poisson Lie 2-groups and Lie 2-bialgebras. In fact, we also prove that there is a one-to-one correspondence between connected, simply connected quasi-Poisson 2-groups and quasi-Lie 2-bialgebras. Our approach relies on a “universal lifting theorem” for Lie 2-groups: an isomorphism between the graded Lie algebras of multiplicative polyvector fields on the Lie 2-group on one hand and of polydifferentials on the corresponding Lie 2-algebra on the other hand.

On the Mean Curvature Evolution of Two-Convex Hypersurfaces

Wed, 01 May 2013 16:04 EDT

John Head

Source: J. Differential Geom., Volume 94, Number 2, 241--266.

Abstract:
We study the mean curvature evolution of smooth, closed, two-convex hypersurfaces in $\mathbb{R}^{n+1}$ for $n \ge 3$. Within this framework we effect a reconciliation between the flow with surgeries—recently constructed by Huisken and Sinestrari in G. Huisken & C. Sinestrari, Mean curvature flow with surgeries of two-convex hypersurfaces ,—and the wellknown weak solution of the level-set flow: we prove that the two solutions agree in an appropriate limit of the surgery parameters and in a precise quantitative sense. Our proof relies on geometric estimates for certain $L^p$-norms of the mean curvature which are of independent interest even in the setting of classicalmean curvature flow. We additionally show how our construction can be used to pass these estimates to limits and produce regularity results for the weak solution.

Growth of Weil-Petersson Volumes and Random Hyperbolic Surface of Large Genus

Wed, 01 May 2013 16:04 EDT

Maryam Mirzakhani

Source: J. Differential Geom., Volume 94, Number 2, 267--300.

Abstract:
In this paper, we investigate the geometric properties of random hyperbolic surfaces of large genus. We describe the relationship between the behavior of lengths of simple closed geodesics on a hyperbolic surface and properties of the moduli space of such surfaces. First, we study the asymptotic behavior of Weil-Petersson volume $V^{g,n}$ of the moduli spaces of hyperbolic surfaces of genus $g$ with $n$ punctures as $g \to \infty$. Then we discuss basic geometric properties of a random hyperbolic surface of genus $g$ with respect to the Weil-Petersson measure as $g \to \infty$.

Classification of Semisimple Symmetric Spaces with Proper-Actions

Wed, 01 May 2013 16:04 EDT

Takayuki Okuda

Source: J. Differential Geom., Volume 94, Number 2, 301--342.

Abstract:
We give a complete classification of irreducible symmetric spaces for which there exist proper $SL(2,\mathbb{R})$-actions as isometries, using the criterion for proper actions by T. Kobayashi and combinatorial techniques of nilpotent orbits. In particular, we classify irreducible symmetric spaces that admit surface groups as discontinuous groups, combining this with Benoist’s theorem.

Tropical Lambda Lengths, Measured Laminations and Convexity

Wed, 01 May 2013 16:04 EDT

R. C. Penner

Source: J. Differential Geom., Volume 94, Number 2, 343--365.

Abstract:
This work uncovers the tropical analogue, for measured laminations, of the convex hull construction in decorated Teichmüller theory; namely, it is a study in coordinates of geometric degeneration to a point of Thurston’s boundary for Teichmüller space. This may offer a paradigm for the extension of the basic cell decomposition of Riemann’s moduli space to other contexts for general moduli spaces of flat connections on a surface. In any case, this discussion drastically simplifies aspects of previous related studies as is explained. Furthermore, a new class of measured laminations relative to an ideal cell decomposition of a surface is discovered in the limit. Finally, the tropical analogue of the convex hull construction in Minkowski space is formulated as an explicit algorithm that serially simplifies a triangulation with respect to a fixed lamination and has its own independent interest.