##### Department of Mathematics,

University of California San Diego

****************************

### Differential Geometry Seminar

## Julie Rowlett

#### UC Santa Barbara

## The Fundamental Gap Conjecture for Triangles

##### Abstract:

The Fundamental Gap Conjecture due to S. T. Yau and M. van de Berg states that for a convex domain in $R^n$ with diameter $d$, the first two positive eigenvalues of the Dirichlet Laplacian satisfy \[\lambda_2 - \lambda_1 \geq \frac{3 \pi^2}{d^2}.\] $\lambda_2 - \lambda_1$ is known as the fundamental gap and has been studied by many authors. It is of natural interest to spectral geometers, and moreover, estimates for the fundamental gap have applications in analysis, statistical mechanics, quantum field theory, and numerical methods. I will discuss joint work with Zhiqin Lu on the fundamental gap when the domain is a Euclidean triangle. Our first result is a compactness theorem for the gap function, which shows that the gap function is unbounded as a triangle collapses to a segment. I will outline our current work which indicates that the equilateral triangle is a strict local minimum for the gap function on triangular domains. Finally, I will discuss how these results combined with numerical methods may be used to prove the well known conjecture that among all triangular domains, the fundamental gap is minimized by the equilateral triangle.

Host: Ben Weinkove

### April 24, 2009

### 4:00 PM

### AP&M 6402

****************************