##### Department of Mathematics,

University of California San Diego

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### UCI-UCR-UCSD Southern California Differential Geometry Seminar

## Yi Wang

#### Johns Hopkins

## Nonuniqueness for a fully nonlinear boundary Yamabe-type problem via bifurcation theory

##### Abstract:

We consider $\sigma_k$-curvature equation with $H_k$-curvature condition on a compact manifold with boundary $(X^{n+1}, M^n, g)$. When restricting to the closure of the positive $k$-cone, this is a fully nonlinear elliptic equation with a fully nonlinear Robin-type boundary condition. We prove a general bifurcation theorem in order to study nonuniqueness of solutions when 2k is less than n. We explicitly give examples of product manifolds with multiple solutions. It is analogous to Schoen example for Yamabe problem on $S^1\times S^{n-1}$. This is joint work with Jeffrey Case and Ana Claudia Moreira.

### October 9, 2018

### 4:00 PM

### UCI- Rowland Hall 306

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