arXiv research

In the product space H^n \times R; we obtain uniform a priori C^0 horizontal length estimates, uniform a priori C^1 boundary gradient estimates, as well as uniform modulus of continuity, for a class of horizontal minimal equations. In two independent variables, we derive a certain uniform global a priori C^1 estimates …

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

We obtain a priori estimates for solutions to the prescribed scalar curvature equation on $S^3$. The usual non-degeneracy assumption on the curvature function is replaced by a new condition, which is necessary and sufficient for the existence of a priori estimates, when the curvature function is a positive Morse functi…

Paper proves estimates for Lagrangian flow singularities.

Study on Kähler metrics with curvature constraints.

Estimates neural network errors for classification problems.

Abstract: Survey on quadratic Hessian equations, their properties, and open problems.

Paper studies unique interior points and estimates for generalized translating soliton problems.

The paper modifies Vafa-Witten equations on 4-manifolds for better solution estimates.

We obtain a maximum principle, and "a priori" upper estimates for solutions of a class of non linear singular elliptic differential inequalities on Riemannian manifolds under the sole geometrical assumption of volume growth conditions. Various applications of the results obtained are presented.

We focus on estimating \emph{a priori} generalization error of two-layer ReLU neural networks (NNs) trained by mean squared error, which only depends on initial parameters and the target function, through the following research line. We first estimate \emph{a priori} generalization error of finite-width two-layer ReLU …

Establishes log-concavity estimates for convex domains' first Dirichlet eigenfunctions.

We construct singular solutions to special Lagrangian equa- tions with subcritical phases and minimal surface systems. A priori estimate breaking families of smooth solutions are also produced cor- respondingly. A priori estimates for special Lagrangian equations with certain convexity are largely known by now.

New method for complex Monge-Ampère equations on Kähler manifolds.

The purpose of this paper is to prove the a priori estimates for constant scalar curvature Kaehler metrics with conic singularities along normal crossing divisors. The zero order estimates are proved by a reformulated version of Alexandrov's maximum principle. The higher order estimates follow from Chen-Cheng's frame …

New estimates for the population risk are established for two-layer neural networks. These estimates are nearly optimal in the sense that the error rates scale in the same way as the Monte Carlo error rates. They are equally effective in the over-parametrized regime when the network size is much larger than the size of…

The paper studies equations on almost Hermitian manifolds with estimates and existence results.

We study families of complex Monge-Ampère equations, focusing on the case where the cohomology classes degenerate to a non big class. We establish uniform a priori $L^{\infty}$-estimates for the normalized solutions, generalizing the recent work of S. Kolodziej and G. Tian. This has interesting consequences in the stud…

The note provides uniform estimates for complex Hessian equations on compact Hermitian manifolds.

Paper establishes estimates for solutions on compact manifolds.

Optimal a priori estimates are derived for the population risk, also known as the generalization error, of a regularized residual network model. An important part of the regularized model is the usage of a new path norm, called the weighted path norm, as the regularization term. The weighted path norm treats the skip c…

Smooth solutions found for modified mean curvature flow in Riemannian manifolds.

Study smooth hypersurfaces with prescribed curvature in Minkowski space.

New method solves complex Monge-Ampère equations on hermitian manifolds.

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

Let $X$ be a compact Kähler manifold and $\om$ a smooth closed form of bidegree $(1,1)$ which is nonnegative and big. We study the classes ${\mathcal E}_χ(X,\om)$ of $\om$-plurisubharmonic functions of finite weighted Monge-Ampère energy. When the weight $χ$ has fast growth at infinity, the corresponding functions are …

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

Estimates prove existence of curvature flow in curved spaces.

Orthogonal matching pursuit (OMP) is a widely used algorithm for recovering sparse high dimensional vectors in linear regression models. The optimal performance of OMP requires \textit{a priori} knowledge of either the sparsity of regression vector or noise statistics. Both these statistics are rarely known \textit{a p…

Paper studies Hessian quotient equations in warped product manifolds.

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

In a series of papers, including the present one, we give a new, shorter proof of Almgren's partial regularity theorem for area minimizing currents in a Riemannian manifold, with a slight improvement on the regularity assumption for the latter. This note establishes a new a priori estimate on the excess measure of an a…

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

Study on special Lagrangian curvature potential equation, proving existence and uniqueness of smooth solutions.

We study a fully nonlinear PDE involving a linear combination of symmetric polynomials of the Kähler form on a Kähler manifold. A $C^0$ \emph{a priori} estimate is proven in general and a gradient estimate is proven in certain cases. Independently, we also provide a method-of-continuity proof via a path of Kähler metri…

We prove a priori estimates for a generalised Monge-Ampère PDE with "non-constant coefficients" thus improving a result of Sun in the Kähler case. We apply this result to the deformed Hermitian Yang-Mills (dHYM) equation of Jacob-Yau to obtain an existence result and a priori estimates for some ranges of the phase angl…

We derive a priori $C^2$ estimates for the $χ$-plurisubharmonic solutions of general complex Hessian equations with right-hand side depending on gradients.

Given a compact Riemannian manifold, with positive Yamabe quotient, not conformally diffeomorphic to the standard sphere, we prove a priori estimates for solutions to the Yamabe problem. We restrict ourselves to the dimensions less than or equal to 7, where the Positive Mass Theorem is known to be true. We also show th…

Paper proves solutions for deformed Hermitian-Yang-Mills equation on almost Hermitian manifolds.

Paper constructs non-symmetric collapsing spacetimes without symmetries.

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

Study elliptic equations on hyperhermitian manifolds with flat hyperkähler metric.

Derives Hessian estimates for Lagrangian mean curvature equation.

This paper is devoted to the regularity analysis of a geodesic equation in the space of Sasakian metrics. Firstly, we reduce the geodesic equation in the space of Sasakian metrics to a Dirichlet problem of degenerate complex Monge-Ampére type eqution on the Kähler cone; secondly, we obtain a priori etimates for the abo…

The paper proves estimates for vortex-type equations on compact Riemann surfaces.

In this paper, we obtain several a-priori estimates for the Calabi flow on projective bundles admitting the generalized Calabi constructions.

We prove an a priori estimate of type sup*inf on Riemannian manifold of dimension 3 (not necessarily compact).

In this paper we perform a fine blow-up analysis for a fourth order elliptic equation involving critical Sobolev exponent, related to the prescription of some conformal invariant on the standard sphere. We derive from this analysis some a priori estimates in dimension 5 and 6. On the five dimensionl sphere these a prio…