In this paper, we proved the existence of n+5/2 prime closed geodesics on a bumpy Finsler sphere (S-n, F) with an odd integer n >= 3 if all geometrically distinct prime closed geodesics have positive Morse indices.
First we construct minimal hypersurfaces M C Rn+1 in a neighborhood of the origin, with an isolated singularity but cylindrical tangent cone C & times; R, for any strictly minimizing strictly stable cone C in R-n. We show that many of these hypersurfaces are area minimizing. Next, we prove a strong unique continuation result for minimal hypersurfaces V with such a cylindrical tangent cone, stating that if the blowups of V centered at the origin approach C & times; R at infinite order, then V = C & times; R in a neighborhood of the origin. Using this we show that for quadratic cones C = C(S-p & times; S-q), in dimensions n > 8, all O(p + 1) & times; O(q + 1)-invariant minimal hypersurfaces with tangent cone C & times; R at the origin are graphs over one of the surfaces that we constructed. In particular such an invariant minimal hypersurface is either equal to C & times; R or has an isolated singularity at the origin.
Let E-tau be the torus with periods 1, tau and delta(0) be the Dirac measure at 0. Consider the following curvature equation (0.1) triangle u + e(u)= 8 pi n delta(0) on E tau, and the integral Lame equation (0.2) y '' = (n(n + 1)& wp;(z | tau) + B)y, where & wp;(z | tau) is the Weierstrass elliptic function. Motivated by our previous works, we conjecture that solvability of (0.1) depends on the geometry of E-tau determined by the multiple Green function G(n) of E-tau. The main purpose of this paper is to confirm our conjecture. Let F-0 be a fundamental domain of Gamma(2) (see (1.4) for the definition). Define LWn = the continuous part of {tau E F-0 | G(n)(z(1), ..., z(n); tau) has a degenerate trivial critical point}. Among other things, we prove the following results. (i) A necessary and sufficient condition for the existence of solutions of (0.1) with tau E F-0, that is, there are n(n+1)/ 2 simply connected open domains Lambda((k)) (j) in F-0 such that (0.1) has an even solution iff tau E U- j,U-k Lambda((k)) (j) and partial derivative Lambda((k))(j)subset of LWn. (ii) If the monodromy matrices S-i, i = 1, 2, of (0.2) are unitarizable, and lambda is an eigenvalue of S-1 or S-2, then lambda is an element of/ {+/- 1}. The is the best possible result for the integral Lame equation (0.2). (iii) For n = 2, 3, 4, we prove the asymptotic behavior of Re tau for those tau where (0.1) has a solution, as tau -> oc. We remark that the paper is a culmination of our previous works over more than ten years.
In arXiv:1911.08213 it was conjectured that the compactly supported cohomology of the m-th restricted contact locus of an isolated hypersurface singularity coincides, up to a shift, with the Floer cohomology of the m-th iterate of the monodromy of the Milnor fiber. In this paper we give an affirmative answer to this conjecture in the case of plane curves.
We prove the Strominger-Yau-Zaslow mirror symmetry conjecture for non-compact Calabi-Yau surfaces arising from, on the one hand, pairs $(\check{Y},\check{D})$ of a del Pezzo surface $\check{Y}$ and $\check{D}$ a smooth anti-canonical divisor and, on the other hand, pairs $(Y,D)$ of a rational elliptic surface $Y$, and $D$ a singular fiber of Kodaira type $I_k$. Three main results are established concerning the latter pairs $(Y,D)$. First, adapting work of Hein, we prove the existence of a complete Calabi-Yau metric on $Y\setminus D$ asymptotic to a (generically non-standard) semi-flat metric in every K\"ahler class. Secondly, we prove an optimal uniqueness theorem to the effect that, modulo automorphisms, every K\"ahler class on $Y\setminus D$ admits a unique asymptotically semi-flat Calabi-Yau metric. This result yields a finite dimensional K\"ahler moduli space of Calabi-Yau metrics on $Y\setminus D$. Further, this result answers a question of Tian-Yau and settles a folklore conjecture of Yau in this setting. Thirdly, we prove that $Y\setminus D$ equipped with an asymptotically semi-flat Calabi-Yau metric $\omega_{CY}$ admits a special Lagrangian fibration whenever the de Rham cohomology class of $\omega_{CY}$ is not topologically obstructed. Combining these results we define a mirror map from the moduli space of del Pezzo pairs $(\check{Y}, \check{D})$ to the complexified K\"ahler moduli of $(Y,D)$ and prove that the special Lagrangian fibration on $(Y,D)$ is $T$-dual to the special Lagrangian fibration on $(\check{Y}, \check{D})$ previously constructed by the authors. We give some applications of these results, including to the study of automorphisms of del Pezzo surfaces fixing an anti-canonical divisor.
We show that on every non-G_2 complex symmetric space of rank two, there are complete Calabi-Yau metrics of Euclidean volume growth with prescribed horospherical singular tangent cone at infinity, providing the first examples of affine Calabi-Yau smoothings of singular and irregular tangent cone. As a corollary, we obtain infinitely many examples of Calabi-Yau manifolds degenerating to the tangent cone in a single step, supporting a recent conjecture by Sun-Zhang, which was only proved when the tangent cone at infinity has only an isolated singularity.
We prove the instability of conformally Kähler, compact or ALF Einstein 4-manifolds with nonnegative scalar curvature which are not half conformally flat. This applies to all the known examples of gravitational instantons which are not hyperKähler and to the Chen-Lebrun-Weber metric in particular.
We show that the volume of transcendental big $(1,1)$-classes on compact K\"ahler manifolds can be realized by convex bodies, thus answering questions of Lazarsfeld-Musta\c{t}\u{a} and Deng. In our approach we use an approximation process by partial Okounkov bodies together with properties of the restricted volume, and we study the extension of K\"ahler currents, as well as the bimeromorphic behavior of currents with analytic singularities. We also establish a connection between transcendental Okounkov bodies and toric degenerations.
In this paper, we investigate the following curvature equation: \begin{equation} \Delta u+e^{u}=8\pi (\delta _{0}+\delta _{\frac{\omega _{k}}{2}})\text{ in } E_{\tau }\text{, }\tau \in \mathbb{H} (0.1) \label{a} \end{equation} Here $E_{\tau }$ represents a flat torus and $\frac{\omega _{k}}{2}$ is one of the half periods of $E_{\tau }$. Our primary objective is to establish a necessary and sufficient criterion for the existence of a non-even family of solutions (see the definition in Section 1). Remarkably, this is equivalent to determining the presence of solutions for the equation with a single conical singularity: \begin{equation*} \Delta u+e^{u}=8\pi \delta _{0}\text{ in }E_{\tau }\text{, }\tau \in \mathbb{ H}\text{.} \end{equation*} This study marks the first exploration of the structure of non-even families of solutions to the curvature equation with multiple singular sources in the literature. Building on our findings, we provide a comprehensive analysis of the solution structure for equation (0.1) for all $\tau $. This analysis is facilitated by Theorem 1.3, which will play a central role in our exploration of cases involving general parameters in the future, such as: \begin{equation*} \Delta u+e^{u}=8\pi n(\delta _{0}+\delta _{\frac{\omega _{k}}{2}})\text{ in } E_{\tau },\text{ }n\in \mathbb{N}\text{.} \end{equation*} As an application, we offer explicit descriptions for solutions to equation (0.1) in the context of both rectangle tori and rhombus tori. See Corollary 1.4 as well as Corollary 1.5.
The index bundle of a family of Dirac operators associated to an instanton on a multi-Taub-NUT space forms a bow representation. We prove that the gauge equivalence classes of solutions of this bow representation are in one-to-one correspondence with the instantons. We also prove that this correspondence establishes an isometry of the bow and instanton moduli spaces.
We prove that the intrinsic geometry of compact cross-sections of any vacuum extremal horizon must admit a Killing vector field. If the cross-sections are two-dimensional spheres, this implies that the most general solution is the extremal Kerr horizon and completes the classification of the associated near-horizon geometries. The same results hold with a cosmological constant. Furthermore, we also deduce that any non-trivial vacuum near-horizon geometry, with a non-positive cosmological constant, must have an SO(2,1) isometry in all dimensions under no symmetry assumptions. We also show that, if the cross-sections are two-dimensional, the horizon Einstein equation is equivalent to a single fourth order PDE for the K\"ahler potential, and that this equation is explicitly solvable on the sphere if the corresponding metric admits a Killing vector.
We study toroidal compactifications of finite volume complex hyperbolic manifolds. We obtain results on the existence or nonexistence of K\"ahler metrics satisfying certain nonpositive curvature properties on these compactifications. Starting from quotients of complex hyperbolic space by deep enough non-uniform arithmetic lattices, we also verify the Shafarevich conjecture for their compactifications, by showing that their universal covers are Stein.
We prove a lower bound on the length of closed geodesics for spherical surfaces with Willmore energy below $6\pi$. The energy threshold is optimal and there is no comparable result for surfaces of higher genus. We also discuss consequences for the injectivity radius.
Let (M, g) be an n-dimensional asymptotically flat Riemannian manifold with nonnegative scalar curvature that admits a noncompact area-minimizing hypersurface Sigma subset of M. In the case where n = 3, O. Chodosh and the first-named author have proven that (M, g) is necessarily isometric to Euclidean space, confirming a conjecture of R. Schoen. In this paper, we extend this result to dimension 3 < n <= 7 provided that Sigma arises as a limit of isoperimetric surfaces. By contrast, we prove that when 3 < n <= 7, there is no such result for general noncompact area-minimizing Sigma subset of M, even when additional assumptions on the stability of Sigma are imposed.
We describe the families of minimal rational curves on any complete symmetric variety, and the corresponding varieties of minimal rational tangents (VMRT). In particular, we prove that these varieties are homogeneous and that for non-exceptional irreducible wonderful varieties, there is a unique family of minimal rational curves, and hence a unique VMRT. We relate these results to the restricted root system of the associated symmetric space. In particular we answer by the negative a question of Hwang: for certain Fano wonderful symmetric varieties, the VMRT has two connected components.
We prove four results towards a description, in terms of the null support function, of the set of isometric embeddings of the hyperbolic plane into Minkowski 3-space. We show that for sufficiently tame null support function, the corresponding entire surface of constant curvature -1 is complete, and for sufficiently sharp null support function, it is incomplete. Our results apply also to entire surfaces whose curvature is merely bounded.
Brinkmann Lorentz manifolds are those admitting an isotropic parallel vector field. We prove geodesic completeness of the compact and also compactly homogeneous Brinkmann spaces. We also prove, partially, that their parallel vector field generates an equicontinuous flow.
We identify all Anosov representations of compact hyperbolic triangle reflection groups into $\mathrm{SL}(3,\mathbb R)$. Specifically, we prove that such a representation is Anosov if and only if it lies in the Hitchin component of the representation space, or it lies in the Barbot component and the product of the three generators of the triangle group has distinct real eigenvalues.
This article is the second of two in which we develop a geometric framework for analysing silent and anisotropic big bang singularities. In the present article, we record geometric conclusions obtained by combining the geometric framework with Einstein's equations. The main features of the results are the following: The assumptions do not involve any symmetry requirements and are weak enough to be consistent with most big bang singularities for which the asymptotic geometry is understood. The framework gives a clear picture of the asymptotic geometry. It also reproduces the Kasner map, conjectured in the physics literature to constitute the essence of the asymptotic dynamics for vacuum solutions to Einstein's equations. When combined with Einstein's equations, the framework yields partial improvements of the assumptions concerning, e.g., the expansion normalised Weingarten map $\mathcal{K}$ (one of the central objects of the framework, defined as the Weingarten map of the leaves of the foliation divided by the mean curvature). For example, the expansion normalised normal derivative of $\mathcal{K}$ can, under suitable assumptions concerning the eigenvalues of $\mathcal{K}$, be demonstrated to decay exponentially and $\mathcal{K}$ can be demonstrated to converge exponentially, even though we initially only impose weighted bounds on these quantities. Finally, the framework gives a unified perspective on the existing results. Moreover, in $3+1$-dimensions, the only parameters necessary to interpret the results are the eigenvalues of $\mathcal{K}$ and an additional scalar function determined by the geometry induced on the leaves of the foliation. In the companion article, we obtain conclusions concerning the asymptotic behaviour of solutions to linear systems of wave equations on the backgrounds consistent with the framework.
