I will talk about the rigidity for a local holomorphic isometric embedding
from ${\BB}^n$ into ${\BB}^{N_1} \times\cdots \times{\BB}^{N_m}$ with
respect to the normalized Bergman metrics. Each component of the map is a
multi-valued holomorphic map between complex Euclidean spaces by Mok's
algebraic extension theorem. By using the method of the holomorphic
continuation and analyzing real analytic subvarieties carefully, we show
that a component is either a constant map or a proper holomorphic map
between balls. Hence the total geodesy of non-constant components follows
from a linearity criterion of Huang. In fact, the rigidity is derived in a
more general setting for a local holomorphic conformal embedding. This is
a joint work with Y. Zhang.

In the 1980s, van den Berg speculated that for all parallelepipeds the gap between the first two Dirichlet eigenvalues is bounded below by a constant. Yau subsequently formulated the fundamental gap conjecture:

For all convex domains in $\R^n$, the gap between the first two Dirichlet eigenvalues is bounded below by $\frac{3 \pi^2}{d^2}$, where $d^2$ is the diameter of the domain.

This talk concerns the spectral gap between Dirichlet eigenvalues of convex domains in $\R^n$, and in particular, the fundamental gap of simplices and triangles. I will discuss recent progress with Z. Lu on the fundamental gap conjecture for triangles and simplices, new connections between Neumann eigenvalues and Dirichlet gaps, and demonstrate a relationship between the fundamental gap and Bakry-Emery geometry. In conclusion, I will offer ideas and open problems.

Eigenvalues of the Laplacian on triangular domains cannot be computed exactly, in general. But the triangles that extremize the first eigenvalue (the fundamental tone of the membrane) often turn out to be equilateral, or degenerate in some way. These special triangles give sharp eigenvalue bounds for the general case.

Among all triangles with fixed diameter, we prove the degenerate acute isosceles triangle minimizes the Neumann fundamental tone. In the other direction, if we fix perimeter (or area) then the equilateral triangle maximizes the Neumann fundamental tone. Our approach involves variational principles and geometric transformations of the domain, and relies on the explicit formulas for eigenfunctions of equilateral triangles and circular sectors. We also prove symmetry/antisymmetry for eigenfunctions of isosceles triangles.

We define a notion of conformal equivalence for discrete surfaces (surfaces composed of euclidean triangles). For example, multiplying the lengths of all edges incident with a single vertex by the same factor is considered to be a conformal change of metric. It turns out that finding a conformally equivalent flat metric on a given discrete surface amounts to minimizing a globally convex functional on the space of all metrics. This functional involves the Lobachevski function (known in the context of computing the volume of hyperbolic tetrahedra). This is not an accident, since surprisingly the whole theory is stongly related to hyperbolic geometry. There are important practical applications of our method to Computer Graphics in the context of texture mapping.