Tobias Lamm

Conservation laws for fourth order systems in four dimensions

Following an approach of the second author (Rivière, 2007) for conformally invariant variational problems in two dimensions, we show in four dimensions the existence of a conservation law for fourth order systems, which includes both intrinsic and extrinsic biharmonic maps. With the help of this conservation law we prove the continuity of weak solutions of this system. Moreover we use the conservation law to derive the existence of a unique global weak solution of the extrinsic biharmonic map flow in the energy space.

Heat Flow for Extrinsic Biharmonic Maps with Small Initial Energy

AbstractLet Mm and Nn↪ℝk be two compact Riemannian manifolds without boundary. We consider the L2 gradient flow for the energy F(u):=

Boundary regularity for polyharmonic maps in the critical dimension

\frac{1}{2}

Estimates for the energy density of critical points of a class of conformally invariant variational problems

∫M|Δu|2. If m≤ 3 or if m= 4 and F(u0) is small, we show that the heat flow for extrinsic biharmonic maps exists for all time, and that the solution subconverges to a smooth extrinsic biharmonic map as time goes to infinity.

Global estimates and energy identities for elliptic systems with antisymmetric potentials

We prove the energy identity for min-max sequences of the Sacks- Uhlenbeck and the biharmonic approximation of harmonic maps from surfaces into general target manifolds. The proof relies on Hopf-differential type es- timates for the two approximations and on estimates for the concentration radius of bubbles.

Quantitative stratification and higher regularity for biharmonic maps

Abstract We consider the Dirichlet problem for intrinsic and extrinsic k-polyharmonic maps from a bounded, smooth domain Ω ⊆ ℝ2k to a compact, smooth Riemannian manifold N ⊆ ℝ l without boundary. For any smooth boundary data, we show that any k-polyharmonic map u ∈ W k,2(Ω, N) is smooth near the boundary ∂Ω.

Quantitative rigidity results for conformal immersions

Abstract We show the existence of a smooth spherical surface minimizing the Willmore functional subject to an area constraint in a compact Riemannian three-manifold, provided the area is small enough. Moreover, we partially classify complete surfaces of Willmore type with positive mean curvature in Riemannian three-manifolds.

The Cauchy problem for Schrödinger flows into Kähler manifolds

Abstract. We show that the energy density of critical points of a class of conformally invariant variational problems with small energy on the unit 2-disk lies in the local Hardy space . As a corollary we obtain a new proof of the energy convexity and uniqueness result for weakly harmonic maps with small energy on B1.