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We consider the level set formulation of the inverse mean curvature flow. We establish a connection to the problem of $p$ -harmonic functions and give a new proof for the existence of weak solutions.

The volume conjecture and four-dimensional hypersurfaces

Oldřich Kowalski (1982)

Commentationes Mathematicae Universitatis Carolinae

The volume preserving mean curvature flow.

Gerhard Huisken (1987)

Journal für die reine und angewandte Mathematik

Theory of semisymmetric conformally flat and biharmonic submanifolds.

Defever, Filip (1999)

Balkan Journal of Geometry and its Applications (BJGA)

Total curvature for knotted surfaces.

N.H. Kuiper, W. III Meeks (1984)

Inventiones mathematicae

Transformations between surfaces in $m a t h b b R^{4}$ with flat normal and/or tangent bundles.

A. Montesinos-Amilibia (2008)

Revista Matemática Iberoamericana

Transformations for hypersurfaces with vanishing Gauss-Kronecker curvature.

Corro, A.V., Ferreira, W., Tenenblat, K. (2005)

Beiträge zur Algebra und Geometrie

Transnormal graph surfaces.

d'Azevedo Breda, A.M., Craveiro de Carvalho, Francisco José (1994)

Beiträge zur Algebra und Geometrie

Transnormal graphs

Craveiro de Carvalho, F.J. (1980)

Portugaliae mathematica

Transnormal partial tubes.

Al-Banawi, Kamal, Carter, Sheila (2005)

Beiträge zur Algebra und Geometrie

Two-dimensional curvature functionals with superquadratic growth

Ernst Kuwert, Tobias Lamm, Yuxiang Li (2015)

Journal of the European Mathematical Society

For two-dimensional, immersed closed surfaces $f : Σ \to ℝ^{n}$ , we study the curvature functionals $ℰ^{p} (f)$ and $𝒲^{p} (f)$ with integrands ${(1 + | A |}^{2})^{p / 2}$ and ${(1 + | H |}^{2})^{p / 2}$ , respectively. Here $A$ is the second fundamental form, $H$ is the mean curvature and we assume $p > 2$ . Our main result asserts that $W^{2, p}$ critical points are smooth in both cases. We also prove a compactness theorem for $𝒲^{p}$ -bounded sequences. In the case of $ℰ^{p}$ this is just Langer’s theorem [16], while for $𝒲^{p}$ we have to impose a bound for the Willmore energy strictly below $8 π$ as an additional condition....

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