# Mathematics Colloquium

## Professor George Kamberov

speaks on

### Shape Invariants of a Surface in $\R^3$. Recovering the Shape of a Surface from the Mean Curvature.

Abstract. A fundamental question in geometry is: what is the minimal set of invariants that determine the shape of an oriented connected surface in $\R^3$? This question arises also in computer vision, in particular in surface recognition.

Bonnet proved that the shape is uniquely determined by the first and second fundamental forms. In the same paper he initiated the study to what extent the shape can be determined only by the first fundamental form and the mean curvature. His results were extended by Cartan and Chern. The theorems of Bonnet, Cartan, and Chern are local and only for surfaces which do not contain umbilic points. Tribuzi and Lawson observed that a closed Riemannian surface admits at most two noncongruent isometric immersions with a given non-constant mean curvature. It is conjectured that the shape of a closed Riemannian surface is uniquely determined by the mean curvature. Ros proved this conjecture for surfaces with positive mean curvature.

We will present a new approach which allowed us to: (i) prove an unique continuation property for immersions with prescribed mean curvature; (ii) describe the properties of the umbilic locus of a surface in $R^3$ which admits another geometrically different isometric immersion preserving the mean curvature; (iii) prove rigidity results for wide classes of closed surfaces; (iv) calculate an upper bound for the dimension of the space of counter-examples to the rigidity conjecture for surfaces of genus one.

The starting point of our approach is to notice that if the isometries $\psi_1$ and $\psi_2$ $:(M^2,g) \rightarrow \R^3$ have the same mean curvature then the shape difference is represented by a holomorphic quadratic differential, $D$, s.t., $D = 0$ if and only if $\psi_1$ and $\psi_2$ are congruent. We prove that if $D\neq 0$ then its isolated zeroes represent the umbilic points of $\psi_1 (M)$ and $\psi_2(M)$. Our rigidity results are based on the analysis of the foliations of curvature lines of the isometries and their relation to the principal stretch foliations of $D$.

Tea. High tea will be served at 3:30pm in the Lounge.
Emmy's. Certain refreshments will be available at the Emmy's after the talk.
Hostess Megan Kerr is the hostess. Please contact Megan (6-1614) if you are interested in having dinner with the speaker.