相关论文: Sigma Models and Minimal Surfaces
I will talk about my recent work with Fernando Marques where we used Almgren-Pitts Min-max Theory to settle some open questions in Geometry: The Willmore conjecture, the Freedman-He-Wang conjecture for links (jointly with Ian Agol), and the…
In the first part of this paper we show that a set $E$ has locally finite $s$-perimeter if and only if it can be approximated in an appropriate sense by smooth open sets. In the second part we prove some elementary properties of local and…
Let (M,g) be an oriented Riemannian manifold of dimension at least 3 and X a vector field on M. We show that the Monge-Amp\`ere differential system (M.A.S.) for X-pseudosoliton hypersurfaces on (M,g) is equivalent to the minimal…
After quick survey of some key results and open questions about the structure of singularities of minimal surfaces, we discuss recent work~\cite{Sim23} on singularities of stable minimal hypersurfaces, including some simplifications of the…
We study the Dirichlet problem for complex Monge-Ampere equations in Hermitian manifolds with general (non-pseudoconvex) boundary. Our main result extends the classical theorem of Caffarelli, Kohn, Nirenberg and Spruck in the flat case. We…
The partial Legendre transform of a non-linear elliptic differential equation is shown to be another non-linear elliptic differential equation. In particular, the partial Legendre transform of the Monge-Amp\`ere equation is another equation…
We prove minimax theorems for lower semicontinuous functions defined on a Hilbert space. The main tool is the theory of $\Phi$-convex functions and sufficient and necessary conditions for the minimax equality to hold for $\Phi$-convex…
Sequences of canonical conservation laws and generalized symmetries for the lattice Boussinesq and the lattice modified Boussinesq systems are successively derived. The interpretation of these symmetries as differential-difference equations…
In this paper, we prove global second derivative estimates for solutions of the Dirichlet problem for the Monge-Ampere equation when the inhomogeneous term is only assumed to be Holder continuous. As a consequence of our approach, we also…
We construct solutions to Monge-Amp\`ere equations whose Monge-Amp\`ere measures contain singular components supported on low codimensional sets. We also study the regularity of such solutions. To motivate our construction, we present…
The WDVV equations of associativity arising in twodimensional topological field theory can be represented, in the simplest nontrivial case, by a single third order equation of the Monge-Ampe`re type. By investigating its Lie point…
We give a survey of various existence results for minimal Lagrangian graphs. We also discuss the mean curvature flow for Lagrangian graphs.
Let $ \Omega \subsetneq \mathbf{R}^n\,(n\geq 2)$ be an unbounded convex domain. We study the minimal surface equation in $\Omega$ with boundary value given by the sum of a linear function and a bounded uniformly continuous function in $…
We discuss a special class of solutions to the minimal surface system. These are vector-valued functions that "decrease area" and are natural generalization of scalar functions. After defining area-decreasing maps, we show several classical…
We give an introduction to our work on the solution to the non-Archimedean Monge-Ampere equation and make comparisons to the complex counterpart. These notes are partially based on talks at the 2015 Simons Symposium on Tropical and…
We construct local zero curvature representations for non-linear sigma models on homogeneous spaces, defined on a space-time of any dimension, following a recently proposed approach to integrable theories in dimensions higher than two. We…
We derive infinitely many conservation laws for some multi-dimensionally consistent lattice equations from their Lax pairs. These lattice equations are the Nijhoff-Quispel-Capel equation, lattice Boussinesq equation, lattice nonlinear…
We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampere type. These are: the problem of locally prescribed Gaussian curvature for surfaces in R^3, and the…
We show here a "weak" H\"older-regularity up to the boundary of the solution to the Dirichlet problem for the complex Monge-Amp\`{e}re equation with data in the $L^p$ space and the boundary of the domain satisfying an $f$-property. The…
We study certain obstacle type problems involving standard and nonlocal minimal surfaces. We obtain optimal regularity of the solution and a characterization of the free boundary.