Student Analysis Seminar (Spring 2026): Mean Curvature Flow

In the first meeting, we will set out the goals of the seminar and discuss various aspects of the references, schedule, and content of the talks.

In the first meeting, we will review some key properties of ancient solutions to mean curvature flow. We will especially focus on asymptotically cylindrical flows and their behavior.

We start our examination of the "PDE-ODI principle" due to Bamler and Lai, cf. Section 4 of https://arxiv.org/pdf/2512.25050 . We will discuss this technique in generality and study its applicability to mean curvature flow and its rescaled counterparts.

We continue our examination of the "PDE-ODI principle" due to Bamler and Lai, cf. Section 4 of https://arxiv.org/pdf/2512.25050 . We will discuss this technique in generality and study its applicability to mean curvature flow and its rescaled counterparts.

The interest around fractional minimal surfaces stems from the non-local nature of various problems involving phase transition, population dynamics, image reconstruction, and many more. For this reason, these mathematical objects have been extensively studied over the past two decades. 

Building on the seminal paper of Caffarelli, Roquejoffre, and Savin, we will discuss existence and Hölder regularity of fractional minimal surfaces, with hints to more recent results concerning convergence to the classical perimeter, higher regularity for minimizers when the fractional parameter s is close to 1, and a generalization of the regularity theory to sets that are only almost minimal.

We will talk about a class of rescaled modified mean curvature flows for which the associated Killing fields are chosen in a way that the neutral mode of the evolving function almost vanishes when we apply the PDE-ODI principle.

This talk will explore the proof of the multiplicity one conjecture due to Bamler and Kleiner. We will study how to establish integral estimates that control the separation of different sheets along the flow.

We continue our examination of the proof of the multiplicity one conjecture due to Bamler and Kleiner. In this lecture, we will study how to establish integral estimates that control the separation of different sheets along the flow.

In recent decades, the existence theory of minimal surfaces has been greatly advanced due to the exciting development of min-max theory, which is an infinite-dimensional Morse theory for the area functional. This culminates in a resolution of Yau’s conjecture (1982) on the existence of infinitely many minimal surfaces in any 3-manifold.

In this talk, I will present a gluing approach to Yau’s conjecture for generic metrics. A key feature of the minimal surfaces obtained is that they have bounded area and unbounded genus. Our work relates Kapouleas-McGrath’s doubling machinery to the variational theory for a Coulomb-type operator for  Schrödinger operators. This is based on a joint work with Daniel Stern.

The smooth mean curvature flow often develops singularities, making weak solutions essential for extending the flow beyond singular times, as well as having applications for geometry and topology. Among various weak formulations, the level set flow method is notable for ensuring long-time existence and uniqueness. However, this comes at the cost of potential fattening, which reflects the genuine non-uniqueness of the flow after singular times. Even for flows starting from smooth, embedded, closed initial data, such non-uniqueness can occur. Thus, we can't expect genuine uniqueness in general. Addressing this non-uniqueness issue is a difficult problem. With Alec Payne, we establish an intersection principle comparing two intersecting flows. We prove that level set flows satisfy this principle in the absence of non-uniqueness.