# Research

I study geometric analysis. For me this has meant fully nonlinear elliptic PDE's such as the special Lagrangian equations, some nonlinear PDE's in conformal geometry, and some PDE's related to the optimal transportation problem.

My family, left to right, Arunima Bhattacharya, Micah Warren, Yu YUAN, Mikhail Safonov, Nikolay Krylov

Papers:

### A flow toward Hamiltonian Stationary Manifolds w/ Jingyi CHEN

We explore a gradient flow for volume on Hamiltonian isotopy classes of Lagrangian submanifolds. The flow is modeled locally by a fourth order quasilinear parabolic equation. We prove short-time existence, and that the flow will be extendable as long as the second fundamental form is bounded. Preprint : here

### Regularity of Hamiltonian Stationary Equations in Symplectic manifolds. w/ Arunima Bhattacharya and Jingyi CHEN.

**To appear Advances in Math. **

In this paper, we prove that any C1-regular Hamiltonian stationary Lagrangian submanifold in a symplectic manifold is smooth. More broadly, we develop a regularity theory for a class of fourth order nonlinear elliptic equations with two distributional derivatives. Our fourth order regularity theory originates in the geometrically motivated variational problem for the volume functional, but should have applications beyond. link to download paper: here

### \C^2,\alpha estimates for solutions to almost linear elliptic equations w/ Arunima Bhattacharya

### Regularity Bootstrapping For Fourth Order Non Linear Elliptic Equations. w/ Arunima Bhattacharya

**Int. Math. Res. Not
**

We consider nonlinear fourth order elliptic equations of double divergence type. We show that for a certain class of equations where the nonlinearity is in the Hessian, solutions that are \(C^{2,a} \) enjoy interior estimates on all derivatives. link to download paper: here

### Compactification of the space of Hamiltonian Stationary Lagrangain Submanifolds of Bounded Total Extrinsic curvature and volume. w/ Jingyi CHEN

### Interior Schauder estimates for the fourth order Hamiltonian stationary equation in two dimensions with Arunima Bhattacharya

### Minimal Lagrangian submanifolds of weighted Kim-McCann metrics.

### Continuum Nash Bargaining Solutions

Nash`s classical bargaining solution suggests that n players in a non-cooperative bargaining situation should find a solution that maximizes the product of each playerss utility functions. We consider a special case: Suppose that the players are chosen from a continuum distribution \( \mu \) and suppose they are to divide up a resource \( \nu \) that is also on a continuum. The utility to each player is determined by the exponential of a distance type function. The maximization problem becomes an optimal transport type problem, where the target density is the minimizer to the functional \( F(\beta)=H_{\nu}(\beta)+W^{2}(\mu,\beta) \) where \(H_{\nu}(\beta) \) is the entropy and \(W^{2}\) is the 2-Wasserstein distance. This minimization problem is also solved in the Jordan-Kinderlehrer-Otto scheme. Thanks to optimal transport theory, the solution may be described by a potential that solves a fourth order nonlinear elliptic PDE, similar to Abreu`s equation. \ Using the PDE, we prove solutions are smooth when the measures have smooth positive densities. \ link to download paper: here

### Radial solutions of a fourth order Hamiltonian stationary equation w/ Jingyi Chen

**Journal of Differential Equations**

We consider smooth radial solutions to the Hamiltonian stationary equation which are defined away from the origin. We show that in dimension two all radial solutions on unbounded domains must be special Lagrangian. In contrast, for all higher dimensions there exist non-special Lagrangian radial solutions over unbounded domains; moreover, near the origin, the gradient graph of such a solution is continuous if and only if the graph is special Lagrangian link to download paper: here

### On a fourth order Hamiltonian stationary equation: Regularity and removable singularities w/ Jingyi Chen

**Advances in Math. **

We prove a Morrey-type theorem for Hamiltonian stationary submanifolds of \( \mathbb{C}^{n} \). Namely, if \(L \subset \mathbb{C}^{n}\) is a \(C^{1}\) Lagrangian submanifold with weakly harmonic Lagrangian phase \( \theta,\) then $L$ must be smooth. In the process we also discuss a local version of the equation, which is a nonlinear fourth order double divergence equation of the potential function whose gradient graph defines the Hamiltonian stationary submanifolds locally, and we establish full regularity and removability of singular sets of capacity zero for weak solutions with \(C^{1,1}\) norm below a dimensional constant. link to download paper: here

### Figure 8 curves that shrink to a point w/ G. Drugan and W. He

** Comm. Anal. Geometry **

We show that figure 8 curves in the plane satisfying a necessary enclosed volume condition together with a two critical point condition an additional symmetry condition must collapse to a point under curve shortening flow. We are able to give upper and lower bounds on the convergence rate of the diameter. The convergence of the curves satisfying the necessary condition is equivalent to the convergence of solutions of Legendrian curve shortening flow in \(R^3\) with a standard contact structure. link to download paper: here

### A Liouville property for gradient graphs and and Bernstein theorem for a fourth order special Lagrangian equation

Manuscripta Mathematica

Abstract: Using an rotation of Yuan, we observe that the gradient graph of any semiconvex function is a Liouville manifold, that is, does not admit bounded harmonic functions. As a corollary, we find that any solution of the fourth order Hamiltonian stationary equation will \[ \Theta \geq \frac{n-2}{2}\pi +\delta \] must be a quadratic link to download paper: here

### Coarse Ricci curvature as a function on \( M \times M \) w/ Tony Ache

Antonio Ache and Micah Warren. ** Results in Math **

Abstract: We use the framework used by Bakry and Emery in their work on logarithmic Sobolev inequalities to define a notion of coarse Ricci curvature on smooth metric measure spaces alternative to the notion proposed by Y. Ollivier. \ This function can be used to recover the Ricci tensor on smooth Riemannian manifolds by the formula \[ \mathrm{Ric}(\gamma^{\prime}\left( 0\right) ,\gamma^{\prime}\left( 0\right) )=\frac{1}{2}\frac{d^{2}}{ds^{2}}\mathrm{Ric}_{\triangle_{g}% }(x,\gamma\left( s\right) ). \]

link to download paper: here

### Non-polynomial entire solutions to $\sigma_{k}$ equations

**Comm. Partial Differential Equations.**

For $2k=n+1$, we exhibit non-polynomial solutions to the Hessian equation \[ \sigma_{k}(D^{2}u)=1 \] on all of $\mathbb{R}^{n}.$.

link to download paper: here

### Approximating coarse Ricci curvature on submanifolds of Euclidean space w/ Tony Ache

** Adv. Geom **
Antonio Ache and Micah Warren.

Abstract: We define approximations of coarse Ricci curvature based on approximations of the Laplace operator, for a scale \( t \). We show that these definition recover the intrinsic Ricci curvature of a submanifold of Euclidean space in the limit.

link to download paper: here

### A Bernstein result and counterexample for entire solutions to Donaldson's equation

**Proc. AMS **

Abstract: We show that convex entire solutions to Donaldson' equation are quadratic, using a result of Weiyong He. We also exhibit entire solutions to the Donaldson equation that are not of the form discussed by He. In the process we discover some non-trivial entire solutions to complex Monge-Amp\`{e}re equations.

link to download paper: here ERRATA: In the last sentence of the paper we comment that the metric in the given example is not flat. This is not true, the manifold is in fact flat.

### Evans-Krylov Estimates for a nonconvex Monge Ampère equation w/ Jeffrey Streets.

Jeffrey Streets and Micah Warren. **Math. Ann. **

Abstract: We establish Evans-Krylov estimates for certain nonconvex fully nonlinear elliptic and parabolic equations by exploiting partial Legendre transformations. The equations under consideration arise in part from the study of the "pluriclosed flow" introduced by the first author and Tian.

link to download paper: here

### Approximate Ricci Curvature with applications to Manifold Learning w/ Tony Ache

Antonio Ache and Micah Warren. ** Advances in Math. **

Abstract: Based on metric measure space notions of Carre du Champ \(\Gamma_2\), we construct a coarse Ricci curvature which converges at appropriate scale to the \(Ric_{\infty} \) on a smooth metric measure space.

link to download paper: here

### On Solutions to Cournot-Nash Equilibria Equations on the Sphere.

Pac. J. Math.

Abstract: We discuss equations associated to Cournot-Nash Equilibria as put forward recently by Blanchet and Carlier. These equations are related to an optimal transport problem in which the source measure is known, but the target measure is part of the problem. The resulting equation is a Monge-Ampère type with possible nonlocal terms. If the cost function is of a particular form, the equation is vulnerable to standard optimal transportation PDE techniques, with some modifications to deal with the new terms. We give some conditions on the problem from which we can conclude that solutions are smooth.

link to download paper: here (No Background check required.)

### Regularity of optimal transport with Euclidean distance squared cost on the embdedded sphere w/ Jun Kitagawa

Jun Kitagawa and Micah Warren. SIAM J. Math. Anal. 44 (4) (2012), 2871-\962887.

Abstract:We give sufficient conditions on initial and target measures supported on the sphere \( S^n \)to ensure the solution to the optimal transport problem with the cost \( |x - y|^2/2 \) is a diffeomorphism

link to download paper: here (No Background check required.)

### Parabolic otimal transport equations on manifolds w/ Young-Heon Kim and Jeffrey Streets

**Calc. Var. Partial Differential Equations** **41** (1-2) (2011) 21-43.

link to download paper: here

### Rectifiability of Optimal Transportation Plans w/ Robert McCann and Brendan Pass.

**Can. J. Math,** **64** (2012) 924--933

link to download paper: here

### Regularity for a log-concave to log-concave mass transfer problem with near Euclidean cost

**Comm. Anal. Geom. ** Volume 19, Number 1, 191\96208, 2011.

link to download paper: here

### A McLean Theorem for the moduli space of Lie solutions to mass transport equations

** Diff. Geom. Appl.** **29** (2011) 816--825.

link to download paper: here

### A boundary value problem for minimal Lagrangian graphs w/ Simon Brendle

**J. Differential Geom. ** 84 (2010) 267-287.

link to download paper: here

### Conformally bending three-manifolds with boundary w/Matthew Gursky and Jeffrey Street

**Ann. Inst. Fourier (Grenoble).** **60** } (6) (2010) 2421--2447.

link to download paper: here

### Pseudo-Riemannian Geometry Calibrates optimal transportation w/ Young-Heon Kim and Robert McCann

**Math. Res. Lett.** **17** (6) (2010) 1183\96 1197.

link to download paper: here

### Existence of Complete conformal metrics of negative Ricci curvature on manifolds with boundary w/ Matthew Gursky and Jeffrey Streets

**Calc. Var. Partial Differential Equations.** **41** (1-2) (2011) 21-43.
link to download paper: here

### Hessian and gradient estimates for three dimensional special Lagrangian equations with large phase w/ Yu Yuan

**American J. Math**{132}, (3) (2010) link to download paper: here

### Hessian estimates for convex solutions to special Lagrangian equations w/ Jingyi Chen and Yu Yuan

**Comm. Pure. Appl. Math.**{62} (4), (2009) 583-595. link to download paper: here

### Hessian estimates for the sigma-2 equation in dimension three w/ Yu Yuan

### Explicit gradient estimates for minimal Lagrangian surfaces of dimension two. w/ Yu Yuan

### A Liouville type theorem for special Lagrangian equations with constraints w/ Yu Yuan

**Comm. Partial Differential Equations.**33(4-6) (2008) 922--932. link to download paper: here

### Hessian estimates for the sigma-2 equation in dimension three (joint work with Yu Yuan).

### Calibrations Associated to Monge-Amp\`ere Equations.

### Thesis under Yu Yuan

Special Lagrangian Equations

University of Washington, 2008.

link to download paper: here