The Willmore Flow with Small Initial Energy

By Ernst Kuwert, R. Schatzle. 2001

250 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1889f36b809361fe166cd39aaf5c968bfeb04f2f

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Computation of geometric partial differential equations and mean curvature flow (2005, methodology): “They were subsequently able to remove the smallness assumption and to prove the existence of a global smooth solution provided that E(X0) ≤ 16π, where Γ0 = X0(S2) (see Kuwert and Schätzle (2004b) and note that our definition differs from theirs by a factor of 2).”
Gradient flow for the Willmore functional (2002, background): “In a forthcoming joint paper the authors have extended the results of Polden to space curves, treating also the case A = 0 with or without a length constraint [5].”
Analysis aspects of Willmore surfaces (2008, methodology): “This result was previously established by Kuwert and Schaetzle in [ KS1 ] for strong solutions : in the case where the surface is already assumed to be smooth.”
Error analysis of a finite element method for the Willmore flow of graphs (2006, methodology): “Using different methods, Kuwert & Scḧ atzle ([11]) obtain global existence of solutions provided that ∫ Γ0 |A◦|2 is sufficiently small, whereA denotes the trace-free part of the second fundamental form.”
Biharmonic map heat flow into manifolds of nonpositive curvature (2004, background): “[15], [16] for the harmonic map heat flow in two dimensions and the Yang-Mills flow in four dimensions or by E. Kuwert and R. Sch¨ atzle [ 10 ], [11] for the Willmore flow.”
The Wulff shape minimizes an anisotropic Willmore functional (2004, background): “In [7, 9] they are able to show that for surfaces of sphere type and initial energy less than or equal to 8 π , Willmore flow converges to a round sphere.”
Geometric flows with rough initial data (2009, background): “In the case that Σ is a sphere Kuwert & Schätzle [13]-[15] showed that if W (f0) ≤ 8π, then the Willmore flow exists for all times and subconverges to a smooth Willmore sphere.”
Optimal rigidity estimates for nearly umbilical surfaces (2005, result): “In [16], the author proves a similar result replacing the L∞ condition by some integral versions of it.”