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- Title
Willmore-type variational problem for foliated hypersurfaces.
- Authors
Rovenski, Vladimir
- Abstract
After Thomas James Willmore, many authors were looking for an immersion of a manifold in Euclidean space or Riemannian manifold, which is the critical point of functionals whose integrands depend on the mean curvature or the norm of the second fundamental form. We study a new Willmore-type variational problem for a foliated hypersurface in Euclidean space. Its general version is the Reilly-type functional, where the integrand depends on elementary symmetric functions of the eigenvalues of the restriction on the leaves of the second fundamental form. We find the 1st and 2nd variations of such functionals and show the conformal invariance of some of them. For a critical hypersurface with a transversally harmonic foliation, we derive the Euler-Lagrange equation and give examples with low-dimensional foliations. We present critical hypersurfaces of revolution and show that they are local minima for special variations of immersion.
- Subjects
HYPERSURFACES; RIEMANNIAN manifolds; CONFORMAL invariants; MATHEMATICAL models; MATHEMATICAL analysis
- Publication
Electronic Research Archive, 2024, Vol 32, Issue 6, p1
- ISSN
2688-1594
- Publication type
Article
- DOI
10.3934/era.2024181