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
Preprint :
here
Abstract: We show the existence of a properly immersed translating solution to
curve diffusion flow in the plane.
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Abstract: We prove a Liouville type result for convex solutions of the Lagrangian mean curvature flow with restricted quadratic growth assumptions
at antiquity on the solutions.
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.
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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.
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Commun. Pure Appl. Anal.
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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.
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J. Differential Geom.
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Proc. Amer. Math. Soc.
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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. \
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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
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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.
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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.
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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
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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
\[
\mathrm{Ric}(\gamma^{\prime}\left( 0\right) ,\gamma^{\prime}\left(
0\right) )=\frac{1}{2}\frac{d^{2}}{ds^{2}}\mathrm{Ric}_{\triangle_{g}%
}(x,\gamma\left( s\right) ).
\]
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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 $\mathbb{R}^{n}.$.
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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.
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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.
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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.
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.
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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.
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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.
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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
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Calc. Var. Partial Differential Equations 41 (1-2) (2011) 21-43.
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Can. J. Math, 64 (2012) 924--933
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Comm. Anal. Geom. Volume 19, Number 1, 191\96208, 2011.
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Diff. Geom. Appl. 29 (2011) 816--825.
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J. Differential Geom. 84 (2010) 267-287.
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Ann. Inst. Fourier (Grenoble). 60 } (6) (2010) 2421--2447.
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Math. Res. Lett. 17 (6) (2010) 1183\96 1197.
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Calc. Var. Partial Differential Equations. 41 (1-2) (2011) 21-43.
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American J. Math {132}, (3) (2010)
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Comm. Pure. Appl. Math. {62} (4), (2009) 583-595.
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Comm. Pure. Appl. Math. {62} (3) (2009) 305-321.
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Math Z. { 262} (4), (2008) pp. 867-879.
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Comm. Partial Differential Equations. 33(4-6) (2008) 922--932.
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Oberwolfach Reports. No. 35 (2007) 2049--2051.
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Trans. AMS. 362}(8), (2010), 3947--3962.
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Special Lagrangian Equations
University of Washington, 2008.
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