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# 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.

### Continuum Nash Bargaining Solutions

Nashs 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 Abreus equation. \ Using the PDE, we prove solutions are smooth when the measures have smooth positive densities. \ link to download paper: here

### Regularity bootstrapping for fourth order nonlinear elliptic equations w/ Arunima Bhattacharya

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

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

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

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

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. http://arxiv.org/abs/1508.01186 link to download paper: here

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

To appear in 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. Preprint.

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 $\mathrm{Ric}(\gamma^{\prime}\left( 0\right) ,\gamma^{\prime}\left( 0\right) )=\frac{1}{2}\frac{d^{2}}{ds^{2}}\mathrm{Ric}_{\triangle_{g}% }(x,\gamma\left( s\right) ).$

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

To appear in 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 $\mathbb{R}^{n}.$.

### Approximating coarse Ricci curvature with applications to submanifolds of Euclidean space w/ Tony Ache

Antonio Ache and Micah Warren. Preprint.

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.

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

To appear in 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.

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

Jeffrey Streets and Micah Warren. To appear in 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.

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

Antonio Ache and Micah Warren. Preprint.

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.

### 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.

### 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

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

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

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

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

### 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.

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

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

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

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

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

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

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

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

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

Comm. Pure. Appl. Math. {62} (3) (2009) 305-321.

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

Math Z. { 262} (4), (2008) pp. 867-879.

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

Oberwolfach Reports. No. 35 (2007) 2049--2051.