About

Panagiota Daskalopoulos is a Professor in the Columbia University Mathematics Department. She earned her Ph.D. from the University of Chicago in 1992. Her research focuses on Partial Differential Equations and Geometric Analysis.

Research topics

• Geometry
• Mathematics
• Physics
• Pure mathematics
• Mathematical analysis

Selected publications

• Classification of bubble-sheet ovals inℝ4

  Geometry & Topology · 2025-04-21 · 5 citations

  articleOpen access
  PublisherOA PDFDOI

• Richard Streit Hamilton (1943–2024)

  Notices of the American Mathematical Society · 2025-10-01

  article
  PublisherDOI

• Mean curvature flow near a peanut solution

  ArXiv.org · 2025-12-04

  preprintOpen access

  It was shown by Angenent, Altschuler and Giga, and by Angenent and Velazquez that there exist closed mean curvature flow solutions that extinct to a point in finite time, without ever becoming convex prior to their extinction. These solutions develop a degenerate neckpinch singularity, meaning that the tangent flow at a singularity is a round cylinder, but at the same time for each of these solutions there exists a sequence of points in space and time, so that the pointed blow up limit around this sequence is the Bowl soliton. These solutions are called peanut solutions and they were first conjectured to exist by Richard Hamilton, while the existence of those solutions was shown by Angenent, Altschuler and Giga. In this paper we show that this type of solutions are highly unstable, in the sense that in every small neighborhood of any such peanut solution we can find a perturbation so that the mean curvature flow starting at that perturbation develops spherical singularity, and at the same time we can find a perturbation so that the mean curvature flow starting at that perturbation develops a nondegenerate neckpinch singularity. We also show that appropriately rescaled subsequence of any sequence of solutions whose initial data converge to the peanut solution, and all of which develop spherical singularities, converges to the Ancient oval solution.

  PublisherOA PDFDOI

• Dynamics of convex mean curvature flow

  Journal für die reine und angewandte Mathematik (Crelles Journal) · 2025-05-02 · 1 citations

  article

  Abstract There is an extensive and growing body of work analyzing convex ancient solutions to mean curvature flow (MCF), or equivalently of rescaled mean curvature flow (RMCF). The goal of this paper is to complement the existing literature, which analyzes ancient solutions one at a time, by considering the space 𝑋 of all convex hypersurfaces <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>M</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:mrow> </m:math> M\subset\mathbb{R}^{n+1} , regard RMCF as a semiflow on this space, and study the dynamics of this semiflow. To this end, we first extend the well-known existence and uniqueness of solutions to MCF with smooth compact convex initial data to include the case of arbitrary noncompact and nonsmooth initial convex hypersurfaces. We identify a suitable weak topology with good compactness properties on the space 𝑋 of convex hypersurfaces and show that RMCF defines a continuous local semiflow on 𝑋 whose fixed points are the shrinking cylinder solitons <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>S</m:mi> <m:mi>k</m:mi> </m:msup> <m:mo lspace="0.222em" rspace="0.222em">×</m:mo> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mi>k</m:mi> </m:mrow> </m:msup> </m:mrow> </m:math> S^{k}\times\mathbb{R}^{n-k} , and for which the Huisken energy is a Lyapunov function. Ancient solutions to MCF are then complete orbits of the RMCF semiflow on 𝑋. We consider the set of all hypersurfaces that lie on an ancient solution that, in backward time, is asymptotic to one of the shrinking cylinder solitons and prove various topological properties of this set. We show that this space is a path connected, compact subset of 𝑋, and considering only point symmetric hypersurfaces, that it is topologically trivial in the sense of Čech cohomology. We also prove that the space of all convex ancient solutions with point symmetry in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> \mathbb{R}^{n+1} is homeomorphic to an <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (n-1) -dimensional simplex, in the case when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> n=2 or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> n=3 , and conjecture that it holds true for any <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> n\geq 2 .

  PublisherDOI

• Uniqueness of ancient solutions to Gauss curvature flow asymptotic to a cylinder

  Journal of Differential Geometry · 2024-05-01 · 8 citations

  articleSenior author

  We address the classification of ancient solutions to the Gauss curvature flow under the assumption that the solutions are contained in a cylinder of bounded cross-section. For each cylinder of convex bounded cross-section, we show that there are only two ancient solutions which are asymptotic to this cylinder: the non-compact translating soliton and the compact oval solution obtained by gluing two translating solitons approaching each other from time $-\infty$ from two opposite ends.

  PublisherDOI

• Dynamics of Convex Mean Curvature Flow

  arXiv (Cornell University) · 2023-05-26

  preprintOpen access

  There is an extensive and growing body of work analyzing convex ancient solutions to Mean Curvature Flow (MCF), or equivalently of Rescaled Mean Curvature Flow (RMCF). The goal of this paper is to complement the existing literature, which analyzes ancient solutions one at a time, by considering the space X of all convex hypersurfaces M, regard RMCF as a semiflow on this space, and study the dynamics of this semiflow. To this end, we first extend the well known existence and uniqueness of solutions to MCF with smooth compact convex initial data to include the case of arbitrary non compact and non smooth initial convex hypersurfaces. We identify a suitable weak topology with good compactness properties on the space X of convex hypersurfaces and show that RMCF defines a continuous local semiflow on X whose fixed points are the shrinking cylinder solitons, and for which the Huisken energy is a Lyapunov function. Ancient solutions to MCF are then complete orbits of the RMCF semiflow on X. We consider the set of all hypersurfaces that lie on an ancient solution that in backward time is asymptotic to one of the shrinking cylinder solitons and prove various topological properties of this set. We show that this space is a path connected, compact subset of X, and, considering only point symmetric hypersurfaces, that it is topologically trivial in the sense of Cech cohomology. We also give a strong evidence in support of the conjecture that the space of all convex ancient solutions with a point symmetry is homeomorphic to an n-1 dimensional simplex.

  PublisherOA PDFDOI

• Uniqueness of entire graphs evolving by mean curvature flow

  Journal für die reine und angewandte Mathematik (Crelles Journal) · 2023-01-27 · 5 citations

  article1st author

  Abstract In this paper we study the uniqueness of graphical mean curvature flow with locally Lipschitz initial data. We first prove that rotationally symmetric entire graphs are unique, without any further assumptions. Our methods also give an alternative simple proof of uniqueness in the one-dimensional case. In the general case, we establish the uniqueness of entire proper graphs that satisfy a uniform lower bound on the second fundamental form. The latter result extends to initial conditions that are proper graphs over subdomains of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> {\mathbb{R}^{n}} . A consequence of our result is the uniqueness of convex entire graphs, which allow us to prove that Hamilton’s Harnack estimate holds for mean curvature flow solutions that are convex entire graphs.

  PublisherDOI

• Convergence of Gauss curvature flows to translating solitons

  Advances in Mathematics · 2022-01-18 · 6 citations

  articleSenior author
  PublisherDOI

• Classification of bubble-sheet ovals in $\mathbb{R}^{4}$

  arXiv (Cornell University) · 2022-09-11

  preprintOpen access

  In this paper, we prove that any bubble-sheet oval for the mean curvature flow in $\mathbb{R}^4$, up to scaling and rigid motion, either is the $\textrm{O}(2)\times \textrm{O}(2)$-symmetric ancient oval constructed by Hershkovits and the fourth author, or belongs to the one-parameter family of $\mathbb{Z}_2^2\times \textrm{O}(2)$-symmetric ancient ovals constructed by the third and fourth author. In particular, this seems to be the first instance of a classification result for geometric flows that are neither cohomogeneity-one nor selfsimilar.

  PublisherOA PDFDOI

• The $$Q_k$$ flow on complete non-compact graphs

  Calculus of Variations and Partial Differential Equations · 2022-02-11 · 8 citations

  preprintOpen accessSenior author
  PublisherOA PDFDOI

Recent grants

• Nonlinear elliptic and parabolic problems in analysis and geometry

  NSF · $178k · 2010–2014

• Nonlinear Geometric Flows: Ancient Solutions, Non-Compact Surfaces, and Regularity

  NSF · $550k · 2019–2026

• Nonlinear Elliptic and Parabolic Problems

  NSF · $140k · 2007–2011

• Nonlinear parabolic equations and related geometric problems

  NSF · $239k · 2013–2017

• Nonlinear Geometric Partial Differential Equations: Entire Solutions and Regularity

  NSF · $299k · 2016–2021

Frequent coauthors

• Nataša Šešum

  55 shared

• Kyeongsu Choi

  25 shared

• Beomjun Choi

  20 shared

• Manuel del Pino

  University of Bath

  18 shared

• Sigurd Angenent

  14 shared

• Ki-Ahm Lee

  14 shared

• Simon Brendle

  11 shared

• Richard S. Hamilton

  7 shared

Education

• Ph.D., Partial Differential Equations and Geometric Analysis

  Chicago

  1992