arXiv research

Estimates for special Lagrangian curvature equations in critical and convex cases.

We establish interior $C^2$ estimates for convex solutions of scalar curvature equation and $σ_2$-Hessian equation. We also prove interior curvature estimate for isometrically immersed hypersurfaces $(M^n,g)\subset \mathbb R^{n+1}$ with positive scalar curvature. These estimates are consequences of an interior estimate…

Study interior estimates for solutions of Poisson equation on Riemann surfaces.

The study provides interior curvature estimates for convex graphs satisfying a specific quotient equation.

Interior $C^{2}$ estimates for sum Hessian quotient equations on Riemannian manifolds

The study provides interior estimates for $Q_k$-flows and translators in $\mathbb{R}^{n+1}$.

In this paper, we consider the Dirichlet problem of a complex Monge-Ampère equation on a ball in $\mathbb C^n$. With $\mathcal C^{1,α}$ (resp. $\mathcal C^{0,α}$) data, we prove an interior $\mathcal C^{1,α}$ (resp. $\mathcal C^{0,α}$) estimate for the solution. These estimates are generalized versions of the Bedford-T…

We prove a priori interior C2 estimate for σ_2 = f in R3, which generalizes Warren-Yuan's result.

We give a maximum principle proof of interior derivative estimates for the Kähler-Ricci flow, assuming local uniform bounds on the metric.

We study the Abreu's equation in n-dimensional polytopes and derive interior estimates of solutions under the assumption of the uniform K-stability.

Sharp estimate for flow in any dimension.

We study a generalized Abreu Equation in $n$-dimensional polytopes and derive interior estimates of solutions under the assumption of the uniform $K$-stability.

In this paper we prove the interior regularity for the solution to the Abreu equation in any dimension assuming the existence of the $C^0$ estimate.

The paper studies fully nonlinear equations on Hermitian manifolds, proving existence and interior estimates.

Paper estimates curvature of convex hypersurfaces with prescribed curvature.

In this paper we consider the evolution of a graph-like hypersurface by anisotropic mean curvature flow, under some restrictions on the anisotropic area integrand. We find interior estimates (in both time and space) on the gradient of such hypersurfaces, depending only on the height of the graph and the anisotropic are…

Let X be a toric surface with Delzant polygon P and u(t) be a solution of the Calabi flow equation on P. Suppose the Calabi flow exists in [0, T). By studying local estimates of the Riemann curvature and the geodesic distance under the Calabi flow, we prove a uniform interior estimate of u(t) for t < T.

We prove a priori interior curvature estimates for hypersurfaces of prescribing scalar curvature equations in dimension three. The method is motivated by the integral method of Warren and Yuan. The new observation here is that the "Lagrangian" submanifold constructed similarly as Harvey and Lawson has bounded mean curv…

New examples show flat singular sets can be arbitrarily complex.

Paper solves Dirichlet problem for $p$-convex hypersurfaces with curvature constraints.

Paper develops estimates for Lagrangian phase changes in 2D.

Paper proves gradient estimates for Lagrangian mean curvature equation.

Solves Dirichlet problem for Lagrangian phase equation with critical and supercritical phase.

New estimate for Curve Shortening Flow improves graphical solutions.

New method solves optimization problems with stochastic objectives and constraints.

We derive a priori interior Hessian estimates for the special Lagrangian equation $σ_{2}=1$ in dimension three.

Study shows stability of Schrödinger operator spectral data on a manifold.

We derive explicit, uniform, a priori interior Hessian and gradient estimates for special Lagrangian equations of all phases in dimension two.

The paper develops various estimates for solutions of a fourth order nonlinear PDE, which corresponds to prescribing the scalar curvature of a toric Kahler metric.

In this paper, we obtain the Bedford-Taylor interior $C^{2}$ estimate and local Calabi $C^{3}$ estimate for the solutions to complex Monge-Ampère equations on Hermitian manifolds.

We derive a priori interior Hessian and gradient estimates for special Lagrangian equation of phase at least a critical value in dimension three.

Local estimates of the maximal curvatures of admissible spacelike hypersurfaces in de Sitter space for k-symmetric curvature functions are obtained. They depend on interior and boundary data.

We extend the interior gradient estimate due to N. Korevaar and L. Simon for solutions of the mean curvature equation from the case of Euclidean graphs to the general case of Killing graphs. Our main application is the proof of existence of Killing graphs with prescribed mean curvature function for continuous boundary …

Derives Hessian estimates for Lagrangian mean curvature equation.

Develops new strategy for Hessian estimates in Lagrangian mean curvature equation.

In this paper, we prove uniform curvature estimates for immersed stable free boundary minimal hypersurfaces which satisfy a uniform area bound. Our result is a natural generalization of the celebrated Schoen-Simon-Yau interior curvature estimates up to the free boundary. A direct corollary of our curvature estimates is…

We derive a priori interior Hessian estimates for special Lagrangian equation with critical and supercritical phases in general higher dimensions. Our unified approach leads to sharper estimates even for the previously known three dimensional and convex solution cases.

We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in $H^n+1$ satisfying $f(κ)=σ\in(0, 1)$ with a prescribed asymptotic boundary $Γ$ at infinity has at least one smooth solution with uniformly bounded hyperbol…

This paper consists of real and complex affine techniques for studying the Abreu equation on toric surfaces. In particular, an interior estimate for Ricci tensor is given.

Develops local elliptic regularity for geometrically-natural operators with low regularity coefficients.

New formula simplifies interior polynomial calculation.

We derive a weighted $L^2$-estimate of the Witten spinor in a complete Riemannian spin manifold $(M^n,g)$ of non-negative scalar curvature which is asymptotically Schwarzschild. The interior geometry of $M$ enters this estimate only via the lowest eigenvalue of the square of the Dirac operator on a conformal compactifi…

We prove a priori interior $C^{2,α}$ estimates for solutions of fully nonlinear elliptic equations of twisted type. For example, our estimates apply to equations of the type convex + concave. These results are particularly well suited to equations arising from elliptic regularization. As application, we obtain a new pr…

The interior polynomial is an invariant of bipartite graphs, and a part of the HOMFLY polynomial of a special alternating link coincides with the interior polynomial of the Seifert graph of the link. We extend the interior polynomial to signed bipartite graphs, and we show that, in the planar case, it is equal to a par…

In this paper, we study the convexity, interior gradient estimate, Liouville type theorem and asymptotic behavior at infinity of translating solutions to mean curvature flow as well as the nonlinear flow by powers of the mean curvature.

Using the convex functions in Grassmannian manifolds we can carry out interior estimates for mean curvature flow of higher codimension. In this way some of the results of Ecker-Huisken can be generalized to higher codimension

The paper proves Hessian estimates for specific geometric flows.

Proves Hessian estimates for special Lagrangian equation with new proofs.