https://neckpinch.com/authors/arunima-bhattacharya/New arXiv papers by Arunima Bhattacharya, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2603.20498arxiv:2603.20498Fri, 20 Mar 2026 00:00:00 GMTArunima Bhattacharya, Micah Warren, Daniel Weser — 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 Kähler geometry, our work shows that techniques from para-Kähler 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.Mean curvature flowLagrangian mean curvature flowRicci-flat & Calabi–YauKähler geometryLimit spaces & RCDLong-time existence & convergence https://arxiv.org/abs/2510.22741arxiv:2510.22741Sun, 26 Oct 2025 00:00:00 GMTArunima Bhattacharya, Ravi Shankar, Jeremy Wall, Diego Yepez — We establish interior estimates for singularities of the Lagrangian mean curvature flow when the Lagrangian phase is critical, i.e., |Θ|≥ (n-2)tfracπ2, and extend our results to the broader class of Lagrangian mean curvature type equations. Our gradient estimates require certain structural conditions, and we construct C^α singular viscosity solutions to show that criticality of the phase is necessary, and that these conditions cannot be removed in dimension one. We also introduce a new method for proving C^(2,α) estimates by exponentiating the arctangent operator into a concave one when |Θ|≥ (n-2)tfracπ2 and n>2.Mean curvature flowLagrangian mean curvature flowSingularitiesWeak & singular flows https://arxiv.org/abs/2407.12756arxiv:2407.12756Wed, 17 Jul 2024 00:00:00 GMTArunima Bhattacharya, Jeremy Wall — In this paper, we prove interior a priori estimates for singularities of the Lagrangian mean curvature flow assuming the Lagrangian phase is supercritical. We prove a Jacobi inequality that holds good when the Lagrangian phase is critical and supercritical. We further extend our results to a broader class of Lagrangian mean curvature type equations.Mean curvature flowLagrangian mean curvature flowSingularities https://arxiv.org/abs/2407.12733arxiv:2407.12733Wed, 17 Jul 2024 00:00:00 GMTArunima Bhattacharya, Micah Warren, Daniel Weser — 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.Mean curvature flowLagrangian mean curvature flowAncient solutions https://arxiv.org/abs/2403.07235arxiv:2403.07235Tue, 12 Mar 2024 00:00:00 GMTArunima Bhattacharya, Jeremy Wall — In this paper, we prove interior Hessian estimates for shrinkers, expanders, translators, and rotators of the Lagrangian mean curvature flow under the assumption that the Lagrangian phase is hypercritical. We further extend our results to a broader class of Lagrangian mean curvature type equations.Mean curvature flowLagrangian mean curvature flowRicci solitonsSelf-shrinkers & translators https://arxiv.org/abs/2211.14257arxiv:2211.14257Fri, 25 Nov 2022 00:00:00 GMTJacob Bernstein, Arunima Bhattacharya — We introduce a family of functionals on submanifolds of Cartan-Hadamard manifolds that generalize the Colding-Minicozzi entropy of submanifolds of Euclidean space. We show that these functionals are monotone under mean curvature flow under natural conditions. As a consequence, we obtain sharp lower bounds on these entropies for certain closed hypersurfaces and observe a novel rigidity phenomenon.Mean curvature flowStability & uniqueness