About

Arunima Bhattacharya is an Assistant Professor in the Department of Mathematics at the University of North Carolina at Chapel Hill. Her research focuses on geometric analysis, particularly on fully nonlinear second and fourth-order elliptic partial differential equations that naturally arise in differential geometry. She studies geometric variational problems by applying tools from minimal surface theory, Lagrangian geometry, Kähler geometry, geometric measure theory, and the theory of elliptic equations. Her work includes the study of area minimization problems among Lagrangian surfaces and the analysis of nonlinear fourth-order elliptic equations associated with volume functional variational problems in the Lagrangian setting. Bhattacharya's academic background includes a B.Sc. from St. Xavier’s College, Kolkata, an M.Sc. from the Tata Institute of Fundamental Research, and a Ph.D. from the University of Oregon. She completed postdoctoral research at the University of Washington and was involved with the Mathematical Sciences Research Institute. Her contributions to the field include advancing the understanding of regularity in Hamiltonian stationary equations, Lagrangian mean curvature equations, and related geometric PDEs.

Research topics

• Geometry
• Mathematical analysis
• Mathematics
• Applied mathematics
• Pure mathematics

Selected publications

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

  ArXiv.org · 2026-03-20

  articleOpen access1st authorCorresponding

  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.

  PublisherOA PDF

• A Liouville type theorem for ancient Lagrangian mean curvature flows

  Communications in Partial Differential Equations · 2025-01-09

  article1st authorCorresponding
  PublisherDOI

• Regularity for Hamiltonian stationary equations in $$\mathbb {R}_{n\le 4}^{n}$$

  Mathematische Zeitschrift · 2025-11-06

  article1st authorCorresponding
  PublisherDOI

• A priori estimates for Singularities of the Lagrangian Mean Curvature Flow with supercritical phase

  Nonlinear Analysis · 2025-05-13 · 2 citations

  articleOpen access1st authorCorresponding

  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.

  PublisherDOI

• The CR-Volume of Horizontal Submanifolds of Spheres

  International Mathematics Research Notices · 2025-02-28

  articleSenior author

  Abstract We study an analog in CR-geometry of the conformal volume of Li–Yau. In particular, to submanifolds of odd-dimensional spheres that are Legendrian or, more generally, horizontal with respect to the sphere’s standard CR-structure, we associate a quantity that is invariant under the CR-automorphisms of the sphere. We apply this concept to a corresponding notion of Willmore energy.

  PublisherDOI

• Variational integrals on Hessian spaces: Partial regularity for critical points

  Nonlinear Analysis · 2025-02-05 · 1 citations

  articleOpen access1st authorCorresponding
  PublisherDOI

• Colding–Minicozzi entropies in Cartan–Hadamard manifolds

  Journal für die reine und angewandte Mathematik (Crelles Journal) · 2025-05-02

  articleSenior author

  Abstract We introduce a family of functionals defined on the set of submanifolds of Cartan–Hadamard manifolds which generalize the Colding–Minicozzi entropy of submanifolds of Euclidean space. We show 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.

  PublisherDOI

• Regularity for Hamiltonian stationary equations in $\mathbb{R}_{n\leq 4}^{n}$

  ArXiv.org · 2025-04-25

  preprintOpen access1st authorCorresponding

  In this paper, we study the regularity of solutions to the Hamiltonian stationary equation in complex Euclidean space. We show that in dimensions $n\leq 4$, for all values of the Lagrangian phase, any $C^{1,1}$ solution is smooth and derive a $C^{k,α}$ estimate for it, where $k \geq 2$.

  PublisherOA PDFDOI

• A Priori Estimates for Singularities of the Lagrangian Mean Curvature Flow with Supercritical Phase

  arXiv (Cornell University) · 2024-07-17 · 1 citations

  preprintOpen access1st authorCorresponding

  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.

  PublisherOA PDFDOI

• Optimal Regularity for Lagrangian Mean Curvature Type Equations

  UNC Libraries · 2024-10-17

  articleOpen access1st authorCorresponding
  PublisherDOI

Frequent coauthors

• Micah Warren

  6 shared

• Ravi Shankar

  6 shared

• Jacob Bernstein

  Johns Hopkins University

  4 shared

• Jeremy Wall

  3 shared

• Jingyi Chen

  University of British Columbia

  2 shared

• Anna Skorobogatova

  Princeton University

  1 shared

• Connor Mooney

  University of California, Irvine

  1 shared

• Daniel Weser

  1 shared

Awards & honors

• Bill Guthridge Fellow