arXiv research

Sharp sub-Gaussian bounds for subsolutions of Trudinger's equation on Riemannian manifolds.

Proves bounded subsolution theorem for complex Monge-Ampère equation on compact Hermitian manifolds.

Uniform bounds for complex equations using Monge-Ampère method.

Paper solves complex Monge-Ampère equation on almost Hermitian manifolds.

Sharp upper bounds found for solutions of a specific equation on Riemannian manifolds.

We study the Dirichlet problem of a class of fully nonlinear elliptic equations on Hermitian manifolds and derive a priori $C^2$ estimates which depend on the initial data on manifolds, the admissible subsolutions and the upper bound of the gradients of the solutions. In some special cases, we obtain the gradient estim…

Finite time for subsolutions on Riemannian manifolds proved.

Researchers solve Dirichlet problem for complex Monge-Ampère equation on Hermitian manifolds.

We provide a self-contained treatment of set-theoretic subsolutions to flow by mean curvature, or, more generally, to flow by mean curvature plus an ambient vector field. The ambient space can be any smooth Riemannian manifold. Most importantly, we show that if two such set-theoretic subsolutions are initially disjoint…

Solves Dirichlet problem for elliptic equations on Hermitian manifolds.

A notion of parabolic C-subsolutions is introduced for parabolic equations, extending the theory of C-subsolutions recently developed by B. Guan and more specifically G. Székelyhidi for elliptic equations. The resulting parabolic theory provides a convenient unified approach for the study of many geometric flows.

We compare various notions of weak subsolutions to degenerate complex Monge-Amp{è}re flows, showing that they all coincide. This allows us to show that the viscosity solution coincides with the envelope of pluripotential subsolutions. Dedicated to Duong Hong Phong on the occasion of his 65th birthday.

We address the restriction problem for viscosity subsolutions of a fully nonlinear PDE on a manifold Z. The constraints on the restrictions of smooth subsolutions to a submanifold X in Z determine a restricted subequation on X. The problem is to show that general (upper semi-continuous) subsolutions restrict to satisfy…

Suppose $f(x,y) + \fracκ{2} \|x\|^2 - \fracσ{2}\|y\|^2$ is convex where $σ>0$, and the argmin function $γ(x) = \{ γ: \inf_y f(x,y) = f(x,γ)\}$ exists and is single valued. We will prove $γ$ is differentiable almost everywhere. As an application we deduce a minimum principle for certain semiconcave subsolutions.

We derive a priori $C^2$ estimates for a class of complex Monge-Ampere type equations on Hermitian manifolds. As an application we solve the Dirichlet problem for these equations under the assumption of existence of a subsolution; the existence result, as well as the second order boundary estimates, is new even for bou…

Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations.

In this paper we characterize the degenerate elliptic equations F(D^2u)=0 whose viscosity subsolutions, (F(D^2u) \geq 0), satisfy the strong maximum principle. We introduce an easily computed function f(t) for t > 0, determined by F, and we show that the strong maximum principle holds depending on whether the integral …

New findings on convexity of special Lagrangian geodesics.

New flow solves LYZ equation on Kähler manifolds.

We study decay and compact support properties of positive and bounded solutions of $Δ_{p} u \geq Λ(u)$ on the exterior of a compact set of a complete manifold with rotationally symmetry. In the same setting, we also give a new characterization of stochastic completeness for the $p$-Laplacian in terms of a global $W^{1,…

Solves Dirichlet problem for specific PSH functions on Hermitian manifolds.

There is an interesting potential theory associated to each degenerate elliptic, fully nonlinear equation $f(D^2u) = 0$. These include all the potential theories attached to calibrated geometries. This paper begins the study of tangents to the subsolutions in these theories, a topic inspired by the results of Kiselman …

Study $\mathrm{C}^2$ estimates for $p$-Hessian equations on closed manifolds.

The paper studies limits of flows on Kähler surfaces, proving convergence to solutions of equations.

Characterizations of entire subsolutions for the 1-harmonic equation of a constant 1$-tension field are given with applications in geometry via transformation group theory. In particular, we prove that every level hypersurface of such a subsolution is calibrated and hence is area-minimizing over$\mathbb{R}$; and every…

Paper aims to minimize ruin probability in insurance companies using Sparre Andersen model.

The paper proves growth estimates for subsolutions of quasilinear equations.

We develop the first steps of a parabolic pluripotential theory in bounded strongly pseudo-convex domains of Cn. We study certain degenerate parabolic complex Monge-Amp{è}re equations, modelled on the K{ä}hler-Ricci flow evolving on complex algebraic varieties with Kawamata log-terminal singularities. Under natural ass…

We revisit the problem of uniqueness for the Ricci flow and give a short, direct proof, based on the consideration of a simple energy quantity, of Hamilton/Chen-Zhu's theorem on the uniqueness of complete solutions of uniformly bounded curvature. With a variation of this quantity and technique, we further prove a uniqu…

Study Kähler-Einstein potentials on stable varieties near singularities

Let $(X,α)$ be a Kähler manifold of dimension n, and let $[ω] \in H^{1,1}(X,\mathbb{R})$. We study the problem of specifying the Lagrangian phase of $ω$ with respect to $α$, which is described by the nonlinear elliptic equation \[ \sum_{i=1}^{n} \arctan(λ_i)= h(x) \] where $λ_i$ are the eigenvalues of $ω$ with respect …

Proves regularity of geodesic equation on Hermitian manifolds.

Study complex Monge-Ampère flows on Kähler manifolds using Perron method.

Smooth solutions found for a curvature problem in hyperbolic space.

The paper proves estimates for solutions to nonlinear equations on manifolds with boundary.

We study a fully nonlinear equation of complex Monge-Ampere type on Hermitian manifolds. We establish the a priori estimates for solutions of the equation up to the second order derivatives with the help of a subsolution.

The paper studies global Yamabe flow on AF manifolds, preserving ADM mass.

The existence of a smooth complete strictly locally convex hypersurface with prescribed scalar curvature and asymptotic boundary at infinity in $\mathbb{H}^{3}$ is proved under the assumption that there exists a strictly locally convex subsolution.

Proves solvability of general inverse σ_k equations with constant coefficients.

Given a smooth positive measure $μ$ on a complete Hermitian manifold with Ricci curvature bounded from below, we prove a pointwise Agmon-type bound for the corresponding Bergman kernel, under rather general conditions involving the coercivity of an associated complex Laplacian on $(0,1)$-forms. Thanks to an appropriate…

Study on existence and properties of continuous solutions to complex Hessian equations.

Theory developed for complex Hessian measures on Hermitian manifolds.

On a manifold with boundary, we deform the metric conformally. This induces a deformation of the Schouten tensor. We fix the metric at the boundary and realize a prescribed value for the product of the eigenvalues of the Schouten tensor in the interior, provided that there exists a subsolution.

We prove weak and strong maximum principles, including a Hopf lemma, for smooth subsolutions to equations defined by linear, second-order, partial differential operators whose principal symbols vanish along a portion of the domain boundary. The boundary regularity property of the smooth subsolutions along this boundary…

The paper establishes inequalities and gradient estimates for harmonic functions on Finsler measure spaces.

We solve the classical Dirichlet problem for a general complex Hessian equation on a small ball in $\bC^n$. Then, we show that there is a continuous solution, in pluripotential theory sense, to the Dirichlet problem on compact Hermitian manifolds with boundary that equipped locally conformal Kähler metrics, provided a …

We obtain $C^2$ a priori estimates for solutions of the nonlinear second-order elliptic equation related to the geometric problem of finding a strictly locally convex hypersurface with prescribed curvature and boundary in a space form. Under the assumption of a strictly locally convex subsolution, we establish existenc…

This paper is devoted to $C^2$ a priori estimates for strictly locally convex radial graphs with prescribed Weingarten curvature and boundary in space forms. By constructing two-step continuity process and applying degree theory arguments, existence results in space forms are established for prescribed Gauss curvature …