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Micah Warren

Publications and source records attributed to Micah Warren.

At least 19 recordsLinked to original sources
arXiv2026

Optimal Transport and Generalized Lagrangian Mean Curvature Flows on Kim-McCann Metrics

We express the mean curvature flow of Lagrangian submanifolds in pseudo-Riemannian manifolds endowed with the Kim-McCann-Warren metric within the framework of generalized mean curvature flow on Kim-McCann manifolds. While generalized mean curvature flow has been studied in K\"ahler geometry, our work shows that techniques from para-K\"ahler geometry arise naturally in the Kim-McCann setting. Using this perspective, we prove that the Lagrangian condition is preserved along the flow. By identifying generalized mean curvature flow with Lagrangian mean curvature flow, we show that the Ma-Trudinger-Wang regularity theory applies to this setting. In particular, the cross-curvature positivity condition of Kim-McCann yields smoothly converging flows of Lagrangian submanifolds. Under the cross-curvature condition, any Lagrangian submanifold avoiding the cut locus converges exponentially to a stationary submanifold, which locally arises as the graph of an optimal transport map. Our framework substantiates the analogy between special Lagrangian geometry in almost Calabi-Yau manifolds and optimal transport theory in the Kim-McCann setting. In particular, we show that Kim-McCann manifolds equipped with a para-holomorphic volume form serve as the natural counterpart to almost Calabi-Yau manifolds.

math.DG↗
arXiv2025

A quantitative stability result for regularity of optimal transport on compact manifolds

We use a Korevaar-style maximum principle approach to show the following: Fixing a $C^{2}$ bound on the log densities of a set of smooth measures, there is a quantifiably-sized Wasserstein neighborhood over which all pairs of such measures will enjoy smooth optimal transport. \ We do this in spite of unhelpful MTW\ curvature, by showing that when the gradient of the Kantorovich potential is small enough, the Hessian ``bound" places the Hessian in one of two disconnected regions, one bounded and the other unbounded. \ Tracking the estimate along a continuity path which starts in the bounded region, we conclude the Hessian must stay bounded.

math.AP↗
arXiv2024

Grim Raindrop: A Translating Solution to Curve Diffusion Flow

We show the existence of a properly immersed translating solution to curve diffusion flow in the plane. Curve diffusion flow is a higher order version of curve shortening flow, namely \[ \left( \frac{dX}{dt}\right) ^{\perp}=-\kappa_{ss}N. \]

math.DG↗
arXiv2024

A Geometric flow towards hamiltonian stationary submanifolds

In this paper, we introduce a geometric flow for Lagrangian submanifolds in a K\"ahler manifold that stays in its initial Hamiltonian isotopy class and is a gradient flow for volume. The stationary solutions are the Hamiltonian stationary Lagrangian submanifolds. The flow is not strictly parabolic but it corresponds to a fourth order strictly parabolic scalar equation in the cotangent bundle of the submanifold via Weinstein's Lagrangian neighborhood theorem. For any compact initial Lagrangian immersion, we establish short-time existence, uniqueness, and higher order estimates when the second fundamental forms are uniformly bounded up to time $T$.

math.DG↗
arXiv2021

Regularity of Hamiltonian Stationary Equations in Symplectic manifolds

In this paper, we prove that any $C^{1}$-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.

math.DG↗
arXiv2020

Approximating coarse Ricci curvature on submanifolds of Euclidean space

For an embedded submanifold $Σ\subset\mathbb{R}^{N}$, Belkin and Niyogi showed that one can approximate the Laplacian operator using heat kernels. Using a definition of coarse Ricci curvature derived by iterating Laplacians, we approximate the coarse Ricci curvature of submanifolds $Σ$ in the same way. For this purpose, we derive asymptotics for the approximation of the Ricci curvature proposed in [AW19]. Specifically, we prove Proposition 3.2 in [AW19].

math.DG↗
arXiv2019

Compactification of the space of Hamiltonian stationary Lagrangian submanifolds with bounded total extrinsic curvature and volume

For a sequence of immersed connected closed Hamiltonian stationary Lagrangian submaniolds in $\mathbb{C}^{n}$ with uniform bounds on their volumes and the total extrinsic curvatures, we prove that a subsequence converges either to a point or to a Hamiltonian stationary Lagrangian $n$-varifold locally uniformly in $C^{k}$ for any nonnegative integer $k$ away from a finite set of points, and the limit is Hamiltonian stationary in ${\mathbb{C}}^{n}$. We also obtain a theorem on extending Hamiltonian stationary Lagrangian submanifolds $L$ across a compact set $N$ of Hausdorff codimension at least 2 that is locally noncollapsing in volumes matching its Hausdorff dimension, provided the mean curvature of $L$ is in $L^{n}$ and a condition on local volume of $L$ near $N$ is satisfied.

math.DG↗
arXiv2017

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 player's utility functions. We consider a special case: Suppose that the players are chosen from a continuum distribution $μ$ and suppose they are to divide up a resource $ν$ 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 \[ F(β)=H_ν(β)+W^{2}(μ,β) \] where $H_ν(β)$ 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.

math.AP↗
arXiv2017

On the Regularity of Hamiltonian Stationary Lagrangian manifolds

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 $θ,$ 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.

math.AP↗
arXiv2016

Radial solutions of a fourth order Hamiltonian stationary equation

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.

math.AP↗
arXiv2015

Coarse Ricci curvature as a function on $M\times M$

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 $$ \mathrm{Ric}(γ^{\prime}\left( 0\right) ,γ^{\prime}\left( 0\right) )=\frac{1}{2}\frac{d^{2}}{ds^{2}}\mathrm{Ric}_{Δ_g}(x,γ\left( s\right) )$$ for any curve $γ(s).$

math.DG↗