PIMS Workshop on Geometric Analysis

The focus will be on harmonic maps and Willmore surfaces.

It should be beneficial to graduate students, PDFs and faculty members in related research areas to hear these talks.

This conference follows the 5-day meeting at BIRS on geometric evolution equations and physics on April 17-22.

Speaker: Tobias Lamm

Title: Small surfaces of Willmore type in Riemannian manifolds

Speaker: Reto Mueller

Title: Ricci flow coupled with harmonic map flow

Abstract: We investigate a coupled system of the Ricci flow on M with the harmonic map flow of a map from M to some closed target manifold N. Surprisingly, the coupled system may be less singular than the Ricci flow or the harmonic map flow alone. In particular, we can always rule out energy concentration of the map a-priori by choosing a coupling constant large enough. Moreover, it suffices to bound the curvature of (M, g(t)) to also obtain control of the evolving map and all its derivatives. Besides these new phenomena, the flow shares many good properties with the Ricci flow. In particular, we can derive the monotonicity of an energy, an entropy and a reduced volume functional.

Speaker: Yuxiang Li

Title: Conformal immersions and its application to Willmore functional

Abstract: We study sequences a conformally immersed, compact Riemann surfaces sequence with fixed genus and bounded Willmore energy. We will show that: 1.If the conformal class sequence converges in moduli space, then the surface sequence will converges in the weakly $W^{2,2}$ sense; 2. If it diverges in moduli space, then the limit of Willmore energies must be greater than or equal to $\min(8\pi,\omega^n_p)$.

As applications, we will imply the existence of minimizer of Willmore energy in a fixed conformal class and give a new proof of the existence of minimizing Willmore surfaces of prescribed genus.