Heat Flow of Extrinsic Biharmonic Maps from a four Dimensional Manifold with Boundary

Abstract

Let (M, g) be a four dimensional compact Riemannian manifold with boundary and (N, h) be a compact Riemannian manifold without boundary. We show the existence of a unique, global weak solution of the heat flow of extrinsic biharmonic maps from M to N under the Dirichlet boundary condition, which is regular with the exception of at most finitely many time slices. We also discuss the behavior of solution near the singular times. As an immediate application, we prove the existence of a smooth extrinsic biharmonic map from M to N under any Dirichlet boundary condition.

Uniqueness of conformal-harmonic maps on locally conformally flat 4-manifolds

“…The corresponding heat flows have been studied extensively as well, as a tool to prove the existence of biharmonic maps and polyharmonic maps in a given homotopy class, see e.g. [26,27,13,45,20,21,32]. It should be noted that the extrinsic and intrinsic cases come in two different flavors: the intrinsic variants are considered more geometrically natural because they do not depend on the embedding of the target manifold N ֒→ R K , although they are less natural from the variational point of view due to the lack of coercivity for the intrinsic energies (and thus they are considerably more difficult analytically and less studied); on the other hand, the extrinsic variants are more natural from the analytical point of view but in turn they do depend on the embedding of N ֒→ R K .…”

Energy convexity of intrinsic bi-harmonic maps and applications I: spherical target

“…where u 0 ∈ W 2,2 (B 1 , N ). There are several results of existence for extrinsic bi-harmonic map heat flow, see for instance Lamm [24] for small initial data and Gastel [11] and Wang [41] for solutions with many finitely singular time and any initial data, see also [17] and [18]. Moreover, the solutions to (8) satisfy the following energy inequality…”