Showing papers in "Journal of Differential Geometry in 2016"

Conic singularities metrics with prescribed Ricci curvature: the case of general cone angles along normal crossing divisors

Abstract: Let $X$ be a non-singular compact Kahler manifold, endowed with an effective divisor $D=\sum{(1-\beta_k) Y_k}$ having simple normal crossing support, and satisfying $\beta_k \in (0, 1)$. The natural objects one has to consider in order to explore the differential-geometric properties of the pair $(X,D)$ are the so-called metrics with conic singularities. In this article, we complete our earlier work concerning the Monge-Ampere equations on $(X,D)$ by establishing Laplacian and $\mathscr{C}^{2,\alpha,\beta}$ estimates for the solution of these equations regardless of the size of the coefficients $0 \lt \beta_k \lt 1$. In particular, we obtain a general theorem concerning the existence and regularity of Kahler–Einstein metrics with conic singularities along a normal crossing divisor.

Compact moduli spaces of Del Pezzo surfaces and Kähler–Einstein metrics

Abstract: We prove that the Gromov–Hausdorff compactification of the moduli space of Kahler–Einstein Del Pezzo surfaces in each degree agrees with certain algebro-geometric compactification. In particular, this recovers Tian’s theorem on the existence of Kahler–Einstein metrics on smooth Del Pezzo surfaces and classifies all the degenerations of such metrics. The proof is based on a combination of both algebraic and differential geometric techniques.

From one Reeb orbit to two

Abstract: We show that every (possibly degenerate) contact form on a closed three-manifold has at least two embedded Reeb orbits. We also show that if there are only finitely many embedded Reeb orbits, then their symplectic actions are not all integer multiples of a single real number; and if there are exactly two embedded Reeb orbits, then the product of their symplectic actions is less than or equal to the contact volume of the manifold. The proofs use a relation between the contact volume and the asymptotics of the amount of symplectic action needed to represent certain classes in embedded contact homology, recently proved by the authors and V. Ramos.

Dar’s conjecture and the log–Brunn–Minkowski inequality

Abstract: In 1999, Dar conjectured that there is a stronger version of the celebrated Brunn-Minkowski inequality. However, as pointed out by Campi, Gardner, and Gronchi in 2011, this problem seems to be open even for planar $o$-symmetric convex bodies. In this paper, we give a positive answer to Dar’s conjecture for all planar convex bodies. We also give the equality condition of this stronger inequality. For planar $o$-symmetric convex bodies, the log–Brunn–Minkowski inequality was established by Boroczky, Lutwak, Yang, and Zhang in 2012. It is stronger than the classical Brunn–Minkowski inequality, for planar $o$-symmetric convex bodies. Gaoyong Zhang asked if there is a general version of this inequality. Fortunately, the solution of Dar’s conjecture, especially, the definition of “dilation position”, inspires us to obtain a general version of the log–Brunn–Minkowski inequality. As expected, this inequality implies the classical Brunn–Minkowski inequality for all planar convex bodies.

Topologically slice knots of smooth concordance order two

Abstract: The existence of topologically slice knots that are of infinite order in the knot concordance group followed from Freedman’s work on topological surgery and Donaldson’s gauge theoretic approach to four-manifolds. Here, as an application of Ozsvath and Szabo’s Heegaard Floer theory, we show the existence of an infinite subgroup of the smooth concordance group generated by topologically slice knots of concordance order two. In addition, no nontrivial element in this subgroup can be represented by a knot with Alexander polynomial one.

Kähler manifolds of semi-negative holomorphic sectional curvature

Abstract: In an earlier work, we investigated some consequences of the existence of a Kahler metric of negative holomorphic sectional curvature on a projective manifold. In the present work, we extend our results to the case of semi-negative (i.e., non-positive) holomorphic sectional curvature. In doing so, we define a new invariant that records the largest codimension of maximal subspaces in the tangent spaces on which the holomorphic sectional curvature vanishes. Using this invariant, we establish lower bounds for the nef dimension and, under certain additional assumptions, for the Kodaira dimension of the manifold. In dimension two, a precise structure theorem is obtained.

On the topology and index of minimal surfaces

Abstract: We show that for an immersed two-sided minimal surface in R 3 , there is a lower bound on the index depending on the genus and number of ends. Using this, we show the nonexistence of an embedded minimal surface in R 3 of index 2, as conjectured by Choe (Cho90). Moreover, we show that the index of an immersed two-sided minimal surface with embedded ends is bounded from above and below by a linear function of the total curvature of the surface.

Gross fibrations, SYZ mirror symmetry, and open Gromov–Witten invariants for toric Calabi–Yau orbifolds

Abstract: For a toric Calabi–Yau (CY) orbifold $\mathcal{X}$ whose underlying toric variety is semi-projective, we construct and study a non-toric Lagrangian torus fibration on $\mathcal{X}$, which we call the Gross fibration. We apply the Strominger–Yau–Zaslow (SYZ) recipe to the Gross fibration of $\mathcal{X}$ to construct its mirror with the instanton corrections coming from genus $0$ open orbifold Gromov–Witten (GW) invariants, which are virtual counts of holomorphic orbi-disks in $\mathcal{X}$ bounded by fibers of the Gross fibration. We explicitly evaluate all these invariants by first proving an open/closed equality and then employing the toric mirror theorem for suitable toric (parital) compactifications of $\mathcal{X}$. Our calculations are then applied to (1) prove a conjecture of Gross-Siebert on a relation between genus $0$ open orbifold GW invariants and mirror maps of $\mathcal{X}$—this is called the open mirror theorem, which leads to an enumerative meaning of mirror maps, and (2) demonstrate how open (orbifold) GW invariants for toric CY orbifolds change under toric crepant resolutions—an open analogue of Ruan’s crepant resolution conjecture.

Number of nodal domains and singular points of eigenfunctions of negatively curved surfaces with an isometric involution

Abstract: We prove two types of nodal results for density $1$ subsequences of an orthonormal basis $\{ \phi_j \}$ of eigenfunctions of the Laplacian on a negatively curved compact surface $(M,g)$. The first result pertains to Riemann surfaces $(M, J, \sigma)$ with an anti-holomorphic involution $\sigma$ such that $M - \mathrm{Fix}(\sigma)$ has more than one component. In any genus $\mathfrak{g}$, there is a $(3\mathfrak{g} - 3)$-dimensional moduli space of such real Riemann surfaces. Our main result is that, for any negatively curved $\sigma$-invariant metric $\mathfrak{g}$ on $M$, the number of nodal domains of the even or odd $\Delta_{\mathfrak{g}}$-eigenfunctions tends to infinity along a density $1$ subsequence. For a generic $\sigma$-invariant negatively curved metric $\mathfrak{g}$, the multiplicity of all eigenvalues equals $1$, and all eigenfunctions are either even or odd, and therefore the result holds for almost any eigenfunction. The analytical part of the proof shows that the number of zeros of even eigenfunctions restricted to $\mathrm{Fix}(\sigma)$, and the number of singular points of odd eigenfunctions on $\mathrm{Fix}(\sigma)$, tend to infinity. This is a quantum ergodic restriction phenomenon. Our second result generalizes this statement to any negatively curved surface $(M,g)$ and to a generic curve $C \subset M$: the number of zeros of eigenfunctions $\phi_j |_C$ tends to infinity. The additional step to obtain a growing number of nodal domains in the $(M, J, \sigma)$ setting is topological. It generalizes an argument of Ghosh, Reznikov, and Sarnak on the modular domain to higher genus.

Double quintic symmetroids, Reye congruences, and their derived equivalence

Abstract: Let $\mathscr{Y}$ be the double cover of the quintic symmetric determinantal hypersurface in $\mathbb{P}^{14}$ We consider Calabi–Yau threefolds $Y$ defined as smooth linear sections of $\mathscr{Y}$ In our previous works, we have shown that these Calabi–Yau threefolds $Y$ are naturally paired with Reye congruence Calabi–Yau threefolds $X$ by the projective duality of $\mathscr{Y}$, and observed that these Calabi–Yau threefolds have several interesting properties from the viewpoint of mirror symmetry and also projective geometry In this paper, we prove the derived equivalence between the linear sections $Y$ of $\mathscr{Y}$ and the corresponding Reye congruences $X$

Two closed geodesics on compact simply connected bumpy Finsler manifolds

Abstract: We prove the existence of at least two distinct closed geodesics on a compact simply connected manifold $M$ with a bumpy and irreversible Finsler metric, when $H^* (M; \mathbf{Q}) \cong T_{d,h+1} (x)$ for some integer $h \geq 2$ and even integer $d \geq 2$. Consequently, together with earlier results on $S^n$, it implies the existence of at least two distinct closed geodesics on every compact simply connected manifold $M$ with a bumpy irreversible Finsler metric.

Localized index and $L^2$-Lefschetz fixed-point formula for orbifolds

Abstract: We study a class of localized indices for the Dirac type operators on a complete Riemannian orbifold, where a discrete group acts properly, co-compactly, and isometrically. These localized indices, generalizing the $L^2$-index of Atiyah, are obtained by taking certain traces of the higher index for the Dirac type operators along conjugacy classes of the discrete group, subject to some trace assumption. Applying the local index technique, we also obtain an $L^2$-version of the Lefschetz fixed-point formulas for orbifolds. These cohomological formulas for the localized indices give rise to a class of refined topological invariants for the quotient orbifold.

Period integrals and the Riemann–Hilbert correspondence

Abstract: A tautological system, introduced in [20][21], arises as a regular holonomic system of partial differential equations that governs the period integrals of a family of complete intersections in a complex manifold X, equipped with a suitable Lie group action. A geometric formula for the holonomic rank of such a system was conjectured in [5], and was verified for the case of projective homogeneous space under an assumption. In this paper, we prove this conjecture in full generality. By means of the Riemann–Hilbert correspondence and Fourier transforms, we also generalize the rank formula to an arbitrary projective manifold with a group action.

Local removable singularity theorems for minimal laminations

Abstract: In this paper we prove a local removable singularity theorem for certain minimal laminations with isolated singularities in a Riemannian three-manifold. This removable singularity theorem is the key result used in our proof that a complete, embedded minimal surface in $\mathbb{R}^3$ with quadratic decay of curvature has finite total curvature.

The Ricci flow on manifolds with boundary

Abstract: We study the short-time existence and regularity of solutions to a boundary value problem for the Ricci–DeTurck equation on a manifold with boundary. Using this, we prove the short-time existence and uniqueness of the Ricci flow prescribing the mean curvature and conformal class of the boundary, with arbitrary initial data. Finally, we establish that under suitable control of the boundary data the flow exists as long as the ambient curvature and the second fundamental form of the boundary remain bounded.

Estimates of the mean field equations with integer singular sources: Non-simple blowup

Abstract: Let $M$ be a compact Riemann surface, $\alpha j \gt -1$, and $h(x)$ a positive $C^2$ function of $M$. In this paper, we consider the following mean field equation: \[ \Delta u (x) + \rho \left( \frac{h(x) e^{u(x)}}{\int_{M} h (x) e^{u(x)}} - \frac{1}{\lvert M \rvert} \right) = 4\pi \sum^{d}_{j=1} \alpha_j \left( \delta_{q_j} - \frac{1}{\lvert M \rvert} \right) \textrm{ in } M \textrm{.} \] We prove that for $\alpha_j \in \mathbb{N}$ and any $\rho \gt \rho_0$, the equation has one solution at least if the Euler characteristic $\chi (M) \leq 0$, where $\rho_0 = \underset{M}{\max} ( 2K - \Delta \mathrm{ln} h + N^*)$, $K$ is the Gaussian curvature, and $N^* = 4\pi \sum^{d}_{j=1} \alpha_j$. This result was proved in [10] when $\alpha_j = 0$. Our proof relies on the bubbling analysis if one of the blowup points is at the vortex $q_j$. In the case where $\alpha_j otin \mathbb{N}$, the sharp estimate of solutions near $q_j$ has been obtained in [11]. However, if $\alpha_j \in \mathbb{N}$, then the phenomena of non-simple blowup might occur. One of our contributions in part 1 is to obtain the sharp estimate for the non-simple blowup phenomena.

Min-max minimal hypersurfaces in non-compact manifolds

Abstract: In this work we prove the existence of embedded closed minimal hypersurfaces in non-compact manifolds containing a bounded open subset with smooth and strictly mean-concave boundary and a natural behavior on the geometry at infinity. For doing this, we develop a modified min-max theory for the area functional following Almgren and Pitts' setting, to produce minimal hypersurfaces with intersecting properties. In particular, we prove that any strictly mean-concave region of a compact Riemannian manifold without boundary intersects a closed minimal hypersurface.

Analytic differential equations and spherical real hypersurfaces

Abstract: We establish an injective correspondence $M\longrightarrow\mathcal E(M)$ between real-analytic nonminimal hypersurfaces $M\subset\mathbb{C}^{2}$, spherical at a generic point, and a class of second order complex ODEs with a meromorphic singularity. We apply this result to the proof of the bound $\mbox{dim}\,\mathfrak{hol}(M,p)\leq 5$ for the infinitesimal automorphism algebra of an \it arbitrary \rm germ $(M,p) ot\sim(S^3,p')$ of a real-analytic Levi nonflat hypersurface $M\subset\mathbb{C}^2$ (the Dimension Conjecture). This bound gives the first proof of the dimension gap $\mbox{dim}\,\mathfrak{hol}(M,p)=\{8,5,4,3,2,1,0\}$ for the dimension of the automorphism algebra of a real-analytic Levi nonflat hypersurface. As another application we obtain a new regularity condition for CR-mappings of nonminimal hypersurfaces, that we call \it Fuchsian type, \rm and prove its optimality for extension of CR-mappings to nonminimal points. \\ We also obtain an existence theorem for solutions of a class of singular complex ODEs (Theorem 3.5).

On the B-twisted topological sigma model and Calabi–Yau geometry

Abstract: We provide a rigorous perturbative quantization of the B-twisted topological sigma model via a first-order quantum field theory on derived mapping space in the formal neighborhood of constant maps. We prove that the first Chern class of the target manifold is the obstruction to the quantization via Batalin–Vilkovisky formalism. When the first Chern class vanishes, i.e. on Calabi–Yau manifolds, the factorization algebra of observables gives rise to the expected topological correlation functions in the B-model. We explain a twisting procedure to generalize to the Landau–Ginzburg case, and show that the resulting topological correlations coincide with Vafa’s residue formula.

Jenkins–Serrin-type results for the Jang equation

Abstract: Let $(M,g, k)$ be an initial data set for the Einstein equations of general relativity. We prove that there exist solutions of the Plateau problem for marginally outer trapped surfaces (MOTSs) that are stable in the sense of MOTSs. This answers a question of G. Galloway and N. O’Murchadha raised in “Some remarks on the size of bodies and black holes,” [Classical Quantum Gravity 25 (2008), no. 10, 105009, 9. MR 2416045] and is an ingredient in the proof of the spacetime positive mass theorem given by L.-H. Huang, D. Lee, R. Schoen, and the first named author. We show that a canonical solution of the Jang equation exists in the complement of the union of all weakly future outer trapped regions in the initial data set with respect to a given end, provided that this complement contains no weakly past outer trapped regions. The graph of this solution relates the area of the horizon to the global geometry of the initial data set in a non-trivial way. We prove the existence of a Scherk-type solution of the Jang equation outside the union of all weakly future or past outer trapped regions in the initial data set. This result is a natural exterior analogue for the Jang equation of the classical Jenkins–Serrin theory. We extend and complement existence theorems for Scherk–type constant mean curvature graphs over polygonal domains in $(M,g)$, where $(M,g)$ is a complete Riemannian surface. We can dispense with the a priori assumptions that a sub solution exists and that $(M,g)$ has particular symmetries. Also, our method generalizes to higher dimensions.

Maximal totally geodesic submanifolds and index of symmetric spaces

Abstract: Let $M$ be an irreducible Riemannian symmetric space. The index $i(M)$ of $M$ is the minimal codimension of a totally geodesic submanifold of $M$. In [1] we proved that $i(M)$ is bounded from below by the rank $\mathrm{rk}(M)$ of $M$, that is, $\mathrm{rk}(M) \leq i(M)$. In this paper we classify all irreducible Riemannian symmetric spaces $M$ for which the equality holds, that is, $\mathrm{rk}(M) = i(M)$. In this context we also obtain an explicit classification of all nonsemisimple maximal totally geodesic submanifolds in irreducible Riemannian symmetric spaces of noncompact type and show that they are closely related to irreducible symmetric $\mathrm{R}$-spaces. We also determine the index of some symmetric spaces and classify the irreducible Riemannian symmetric spaces of noncompact type with $i(M) \in \{4, 5, 6 \}$.

Controlling area blow-up in minimal or bounded mean curvature varieties

Abstract: Consider a sequence of minimal varieties $M_i$ in a Riemannian manifold $N$ such that the measures of the boundaries are uniformly bounded on compact sets. Let $Z$ be the set of points at which the areas of the $M_i$ blow up. We prove that $Z$ behaves in some ways like a minimal variety without boundary. In particular, it satisfies the same maximum and barrier principles that a smooth minimal submanifold satisfies. For suitable open subsets $W$ of $N$, this allows one to show that if the areas of the $M_i$ are uniformly bounded on compact subsets of $W$, then the areas are in fact uniformly bounded on all compact subsets of $N$. Similar results are proved for varieties with bounded mean curvature. The results about area blow-up sets are used to show that the Allard Regularity Theorems can be applied in some situations where key hypotheses appear to be missing. In particular, we prove a version of the Allard Boundary Regularity Theorem that does not require any area bounds. For example, we prove that if a sequence of smooth minimal submanifolds converge as sets to a subset of a smooth, connected, properly embedded manifold with nonempty boundary, and if the convergence of the boundaries is smooth, then the convergence is smooth everywhere.

On the volume growth of Kähler manifolds with nonnegative bisectional curvature

Abstract: Let $M$ be a complete Kahler manifold with nonnegative bisectional curvature. Suppose the universal cover does not split and $M$ admits a nonconstant holomorphic function with polynomial growth; we prove $M$ must be of maximal volume growth. This confirms a conjecture of Ni in “A monotonicity formula on complete Kahler manifolds with nonnegative bisectional curvature”, [J. Amer. Math. Soc. 17 (2004), 909–946, MR 2083471, Zbl 1071.58020]. There are two essential ingredients in the proof: the Cheeger–Colding theory on Gromov–Hausdorff convergence of manifolds, and the three-circle theorem for holomorphic functions in “Three circle theorems on Kahler manifolds and applications” by G. Liu [Arxiv: 1308.0710].

Small eigenvalues of closed surfaces

Abstract: Generalizing recent work of Otal and Rosas, we show that the Laplacian of a Riemannian metric on a closed surface $S$ with Euler characteristic $\chi(S) \lt 0$ has at most $-\chi(S)$ small eigenvalues.

Boundary effect of Ricci curvature

Abstract: On a compact Riemannian manifold with boundary, we study how Ricci curvature of the interior affects the geometry of the boundary. First, we establish integral inequalities for functions defined solely on the boundary and apply them to obtain geometric inequalities involving the total mean curvature. Then, we discuss related rigidity questions and prove Ricci curvature rigidity results for manifolds with boundary.

Cohomology and hodge theory on symplectic manifolds: iii

Abstract: arXiv:1402.0427v2 [math.SG] 2 May 2014 Cohomology and Hodge Theory on Symplectic Manifolds: III Chung-Jun Tsai, Li-Sheng Tseng and Shing-Tung Yau February 3, 2014 Abstract We introduce filtered cohomologies of differential forms on symplectic manifolds. They generalize and include the cohomologies discussed in Paper I and II as a subset. The filtered cohomologies are finite-dimensional and can be associated with differential elliptic complexes. Algebraically, we show that the filtered cohomologies give a two-sided resolution of Lefschetz maps, and thereby, they are directly related to the kernels and cokernels of the Lefschetz maps. We also introduce a novel, non-associative product operation on differential forms for symplectic manifolds. This product generates an A ∞ -algebra structure on forms that underlies the filtered cohomologies and gives them a ring structure. As an application, we demonstrate how the ring structure of the filtered cohomologies can distinguish different symplectic four-manifolds in the context of a circle times a fibered three-manifold. Contents 1 Introduction 2 Preliminaries Operations on differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Filtered forms and differential operators . . . . . . . . . . . . . . . . . . . . . . . 14 Short exact sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Foliated stratified spaces and a De Rham complex describing intersection space cohomology

Abstract: The method of intersection spaces associates cell-complexes depending on a perversity to certain types of stratified pseudomanifolds in such a way that Poincare duality holds between the ordinary rational cohomology groups of the cell-complexes associated to complementary perversities. The cohomology of these intersection spaces defines a cohomology theory HI for singular spaces, which is not isomorphic to intersection cohomology IH. Mirror symmetry tends to interchange IH and HI. The theory IH can be tied to type IIA string theory, while HI can be tied to IIB theory. For pseudomanifolds with stratification depth 1 and flat link bundles, the present paper provides a de Rham-theoretic description of the theory HI by a complex of global smooth differential forms on the top stratum. We prove that the wedge product of forms introduces a perversity-internal cup product on HI, for every perversity. Flat link bundles arise for example in foliated stratified spaces and in reductive Borel–Serre compactifications of locally symmetric spaces. A precise topological definition of the notion of a stratified foliation is given.

Topological Type of Limit Laminations of Embedded Minimal Disks

Abstract: We consider two natural classes of minimal laminations in threemanifolds. Both classes may be thought of as limits—in different senses—of embedded minimal disks. In both cases, we prove that, under a natural geometric assumption on the three-manifold, the leaves of these laminations have genus zero. This answers a question posed by Hoffman and White.

Bifurcation of periodic solutions to the singular Yamabe problem on spheres

Abstract: We obtain uncountably many periodic solutions to the singular Yamabe problem on a round sphere that blow up along a great circle. These are (complete) constant scalar curvature metrics on the complement of $\mathbb{S}^1$ inside $\mathbb{S}^m , m \geq 5$, that are conformal to the round (incomplete) metric and periodic in the sense of being invariant under a discrete group of conformal transformations. These solutions come from bifurcating branches of constant scalar curvature metrics on compact quotients of $\mathbb{S}^m \setminus \mathbb{S}^1 \cong \mathbb{S}^{m-2} \times \mathbb{H}^2$.

Affine rigidity of Levi degenerate tube hypersurfaces

Abstract: Let $\mathfrak{C}_{2,1}$ be the class of connected 5-dimensional CR-hypersurfaces that are 2-nondegenerate and uniformly Levi degenerate of rank 1. In our recent article, we proved that the CR-structures in $\mathfrak{C}_{2,1}$ are reducible to $\mathfrak{so}(3, 2)$-valued absolute parallelisms. In the present paper, we apply this result to study tube hypersurfaces in $\mathbb{C}^3$ that belong to $\mathfrak{C}_{2,1}$ and whose CR-curvature identically vanishes. By explicitly solving the zero CR-curvature equations up to affine equivalence, we show that every such hypersurface is affinely equivalent to an open subset of the tube $M_0$ over the future light cone $\lbrace (x_1, x_2, x_3) \in \mathbb{R}^3 \: \vert \: x^2_1 + x^2_2 - x^2_3= 0 , x_3 \gt 0 \rbrace$. Thus, if a tube hypersurface in the class $\mathfrak{C}_{2,1}$ locally looks like a piece of $M_0$ from the point of view of CR-geometry, then from the point of view of affine geometry it (globally) looks like a piece of $M_0$ as well. This rigidity result is in stark contrast to the Levi nondegenerate case, where the CR-geometric and affine-geometric classifications significantly differ.