arXiv research

The symplectic cone of a closed oriented 4-manifold is the set of cohomology classes represented by symplectic forms. A well-known conjecture describes this cone for every minimal Kaehler surface. We consider the case of the elliptic surfaces E(n) and focus on a slightly weaker conjecture for the closure of the symplec…

The study finds billiard trajectories with infinitely many reflections in certain cones.

This paper studies symplectic structures on elliptic surfaces with positive Euler number.

Proves approximation and interpolation for regular immersions directed by algebraically elliptic cones.

In this note we introduce the notion of the relative symplectic cone. As an application, we determine the symplectic cone of certain T^2-fibrations. In particular, for some elliptic surfaces we verify a conjecture on the symplectic cone of minimal Kaehler surfaces raised by the second author.

Characterizes elliptic operators on singular foliations.

We study a class of fully nonlinear elliptic equations on closed Hermitian manifolds. Under the assumption of cone condition, we derive the $L^\infty$ estimate directly.

We generalize McShane's identity for the length series of simple closed geodesics on a cusped hyperbolic surface to hyperbolic cone-surfaces (with all cone angles $\le π$), possibly with cusps and/or geodesic boundary. In particular, by applying the generalized identity to the orbifolds obtained from taking the quotien…

The paper classifies periodic solitons in curve flows on the light-cone.

This is the first paper in a series to develop a linear and nonlinear theory for elliptic and parabolic equations on Kähler varieties with mild singularities. Donaldson has established a Schauder estimate for linear and complex Monge-Ampère equations when the background Kähler metrics on $\mathbb{C}^n$ have cone singul…

In this paper, we study the motion of level sets by general curvature. The difficulty of this setting is that a general curvature function is only well defined in an admissible cone. In order to extend the existence of a weak solution of a general curvature flow to outside the cone we introduce a new approximation func…

Constructs currents and heights on K3 surfaces.

Proof shows cones minimize certain geometric functionals.

We classify all regular three-dimensional convex cones which possess an automorphism group of dimension at least two, and provide analytic expressions for the complete hyperbolic affine spheres which are asymptotic to the boundaries of these cones. The affine spheres are represented by explicit hypersurface immersions …

Researchers compute the index of a specific operator on contact manifolds.

Recently, Atiyah and LeBrun proved versions of the Gauss-Bonnet and Hirzebruch signature Theorems for metrics with edge-cone singularities in dimension four, which they applied to obtain an inequality of Hitchin-Thorpe type for Einstein edge-cone metrics. Interestingly, many natural examples of edge-cone metrics in dim…

Characterizes stably elliptic elements in Lie groups and their properties.

Study on deformations of Spin(7)-structures on manifolds.

The aim of this paper is to investigate uniqueness of conic constant scalar curvature Kaehler (cscK) metrics, when the cone angle is less than $π$. We introduce a new Hölder space called $\cC^{4,\a,\b}$ to study the regularities of this fourth order elliptic equation, and prove that any $\cC^{2,\a,\b}$ conic cscK metri…

We use a weighted variant of the frequency functions introduced by Almgren to prove sharp asymptotic estimates for almost eigenfunctions of the drift Laplacian associated to the Gaussian weight on an asymptotically conical end. As a consequence, we obtain a purely elliptic proof of a result of L. Wang on the uniqueness…

The deformation theory of hyperbolic and Euclidean cone-manifolds with all cone angles less then 2π plays an important role in many problems in low dimensional topology and in the geometrization of 3-manifolds. Furthermore, various old conjectures dating back to Stoker about the moduli of convex hyperbolic and Euclidea…

We give a natural way to identify between two scales, potentially arbitrarily far apart, in a non-compact Ricci-flat manifold with Euclidean volume growth when a tangent cone at infinity has smooth cross section. The identification map is given as the gradient flow of a solution to an elliptic equation.

We study the real Monge-Ampère equation in two and three dimensions, both from the point of view of the SYZ conjecture, where solutions give rise to semi-flat Calabi-Yau's and in affine differential geometry, where solutions yield parabolic affine sphere hypersurfaces. We find explicit examples, connect the holomorphic…

Using an approach based on the heat kernel we prove an Atiyah-Bott-Lefschetz theorem for the $L^2-$Lefschetz numbers associated to an elliptic complex of cone differential operators over a compact manifold with conical singularities. We then apply our results to the case of the de Rham complex.

We study the problem of removable singularities for degenerate elliptic equations. Let F be a fully nonlinear second-order partial differential subequation of degenerate elliptic type on a manifold X. We study the question: Which closed subsets E in X have the property that every F-subharmonic function (subsolution) on…

This is the continuation of our paper \cite{GS}, to study the linear theory for equations with conical singularities. We derive interior Schauder estimates for linear elliptic and parabolic equations with a background Kähler metric of conical singularities along a divisor of simple normal crossings. As an application, …

Researchers found solutions to a complex equation on spheres, overcoming a key difficulty.

Study finds existence and non-uniqueness of cone spherical metrics on compact Riemann surfaces.

New tools for studying Hsiang algebras discovered, linking them to known algebraic structures.

The study explores special hypersurfaces in Riemannian products, focusing on elliptic Weingarten conditions.

We solve the Fu-Yau equation for arbitrary dimension and arbitrary slope $α'$. Actually we obtain at the same time a solution of the open case $α'>0$, an improved solution of the known case $α'<0$, and solutions for a family of Hessian equations which includes the Fu-Yau equation as a special case. The method is based …

This paper is a self-contained presentation of certain aspects of the theory of weighted Sobolev spaces and elliptic operators on non-compact Riemannian manifolds. Specifically, we discuss (i) the standard and weighted Sobolev Embedding Theorems for general manifolds and (ii) Fredholm results for elliptic operators on …

In this paper we develop a systematic deformation theory for conic constant curvature metrics on a closed surface when all cone angles are less than $2π$; in particular, we define and study the Teichmüller space $\mathcal{T}^{\mathrm{conic}}_{γ,k}$ of conic constant curvature metrics on a surface of genus $γ$ with $k$ …

In this paper, we prove the existence of classical solutions of the Dirichlet problem for a class of quasi-linear elliptic equations on unbounded domains like a cone or a U-type domain. This problem comes from the study of mean curvature flow and its generalization, the flow by powers of mean curvature. Our approach is…

Study light ray transform in pseudo-Euclidean space, derive inversion formula, and prove stability.

Study on existence of metrics in conformal geometry with constraints on Schouten tensor.

Bernstein theorem proven for 2-valued minimal graphs in 4D.

New method constructs translationally equivariant hyperbolic affine spheres.

We identify Melrose's suspended algebra of pseudodifferential operators with a subalgebra of the algebra of parametric pseudodifferential operators with parameter space $\R$. For a general algebra of parametric pseudodifferential operators, where the parameter space may now be a cone $Γ\subset\R^p$, we construct a uniq…

Motivated by the picture of mirror symmetry suggested by Strominger, Yau and Zaslow, we made a conjecture concerning the Gromov-Hausdorff limits of Calabi-Yau n-folds (with Ricci-flat Kähler metric) as one approaches a large complex structure limit point in moduli; a similar conjecture was made independently by Kontsev…

One way to generalize the boundary Yamabe problem posed by Escobar is to ask if a given metric on a compact manifold with boundary can be conformally deformed to have vanishing $σ_k$-curvature in the interior and constant $H_k$-curvature on the boundary. When restricting to the closure of the positive $k$-cone, this is…

Introduces holed cone structures to generalize cone structures on 3-manifolds.

Unique cylindrical tangent cone for Simons' hypersurface found.

This is a continuation of the previous articles on Kahler cone metrics. In this article, we introduce weighted function spaces and provide a self-contained treatment on cone angles in the whole interval $(0,1]$. We first construct geodesics in the space of Kahler cone metrics (cone geodesics). We next determine the ver…

Given a closed orientable Euclidean cone 3-manifold C with cone angles less than or equal to pi, and which is not almost product, we describe the space of constant curvature cone structures on C with cone angles less than pi. We establish a regeneration result for such Euclidean cone manifolds into spherical or hyperbo…

Study strict stability of cones with isolated singularities.

Constructs Lie groupoid integrating elliptic tangent bundles and Poisson structures.

Study finds negatively curved spheres in elliptic surfaces and their modifications.