# Geometric Analysis and Topology Seminar

#### Branched regularity theorems for codimension one stable varifolds

**Speaker:**
Paul Minter, IAS

**Location:**
Warren Weaver Hall 512

**Videoconference link:** https://nyu.zoom.us/j/95759976381

**Date:**
Friday, February 17, 2023, 11 a.m.

**Synopsis:**

Abstract: In geometric measure theory, stationary integral varifolds are measure-theoretically defined minimal submanifolds which can be used, amongst other things, to understand limits of smooth minimal submanifolds under uniform volume bounds. However, they can be very singular and so one is led to the problem of understanding their singularities.

In codimension one, the work of N. Wickramasekera provides a very strong understanding of the regularity theory when the varifolds are also “embedded” (in some precise sense) and stable (second variation is non-negative); in particular, they are smoothly embedded outside a set of codimension at least 7. In this setting, the key ingredient to the regularity theory is ruling out a highly-degenerate singularity known as a branch point, which arise in the presence of (integer) multiplicity >1.

In this talk, I will discuss some results in the “immersed” setting, and with no restriction on the multiplicity. The key difference is that branch points can (and do) occur, and indeed the set of branch points has no (non-trivial) a priori bound on its size; indeed, our results allow for it to even have positive measure. We prove that a multi-valued branched C^{1,a} graph structure holds locally about certain singular points, in particular proving uniqueness of tangent cones with quadratic decay to the unique cone. A key aspect of our work is to prove that Almgren’s frequency function is monotone for the multi-valued blow-ups of varifolds in our setting, despite a priori not knowing (in contrast to the area-minimising case) that the blow-ups satisfy any variational property. One application of these results is to understanding branch points in area-minimising hypersurfaces mod p.

Some results are joint with Neshan Wickramasekera.