Willmore-type variational problem for foliated hypersurfaces

Vladimir Rovenski; Vladimir Rovenski

doi:10.3934/era.2024181

Electronic Research Archive

2024, Volume 32, Issue 6: 4025-4042. doi: 10.3934/era.2024181

Previous Article Next Article

Research article Special Issues

Willmore-type variational problem for foliated hypersurfaces

Vladimir Rovenski ^,

Department of Mathematics, University of Haifa, Haifa 3498838, Israel

Received: 29 February 2024 Revised: 08 May 2024 Accepted: 18 June 2024 Published: 21 June 2024

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.
- hypersurface,
- foliation,
- second fundamental form,
- mean curvature,
- Willmore's type functional,
- conformal invariant
Citation: Vladimir Rovenski. Willmore-type variational problem for foliated hypersurfaces[J]. Electronic Research Archive, 2024, 32(6): 4025-4042. doi: 10.3934/era.2024181

Related Papers:

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.

References

[1]	B. Chen, On a theorem of Fenchel-Borsuk-Willmore-Chern-Lashof, Math. Ann., 194 (1971), 19–26. https://doi.org/10.1007/BF01351818 doi: 10.1007/BF01351818
[2]	B. Chen, On the total curvature of immersed manifolds, II. Mean curvature and length of second fundamental form, Am. J. Math., 94 (1972), 799–809. https://doi.org/10.2307/2373759 doi: 10.2307/2373759
[3]	A. Gruber, M. Toda, H. Tran, On the variation of curvature functionals in a space form with application to a generalized Willmore energy, Ann. Glob. Anal. Geom., 56 (2019), 147–165. https://doi.org/10.1007/s10455-019-09661-0 doi: 10.1007/s10455-019-09661-0
[4]	F. C. Marques, A. Neves, Min-max theory and the Willmore conjecture, Ann. Math., 179 (2014), 683–782. https://doi.org/10.4007/annals.2014.179.2.6 doi: 10.4007/annals.2014.179.2.6
[5]	Y. Zhu, J. Liu, G. Wu, Gap phenomenon of an abstract Willmore type functional of hypersurface in unit sphere, Sci. World J., (2014), 697132. https://doi.org/10.1155/2014/697132
[6]	Y. Chang, Willmore surfaces and $F$-Willmore surfaces in space forms, Taiwanese J. Math., 17 (2013), 109–131. http://doi.org/10.11650/tjm.17.2013.1840 doi: 10.11650/tjm.17.2013.1840
[7]	Z. Guo, Higher order Willmore hypersurfaces in Euclidean space, Acta. Math. Sin., Engl. Ser., 25 (2009), 77–84. https://doi.org/10.1007/s10114-008-6422-y doi: 10.1007/s10114-008-6422-y
[8]	J. Li, Z. Guo, Higher order Willmore revolution hypersurfaes in $\mathbb{R}^n$, Lifelong Educ., 10 (2021) 75–89.
[9]	Z. Guo, Generalized Willmore functionals and related variational problems, Differ. Geom. Appl., 25 (2007), 543–551. https://doi.org/10.1016/j.difgeo.2007.06.004 doi: 10.1016/j.difgeo.2007.06.004
[10]	T. J. Willmore, Note on embedded surfaces, An. Sti. Univ. Al. I. Cuza Iasi, N. Ser., 11 (1965), 493–496.
[11]	A. Song, Generation of tubular and membranous shape textures with curvature functionals, J. Math. Imaging Vis., 64 (2022), 17–40. https://doi.org/10.1007/s10851-021-01049-9 doi: 10.1007/s10851-021-01049-9
[12]	R. C. Reilly, Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J. Differ. Geom., 8 (1973), 465–477. https://doi.org/10.4310/jdg/1214431802 doi: 10.4310/jdg/1214431802
[13]	V. Rovenski, P. Walczak, Extrinsic Geometry of Foliations, Springer, Cham, 2021. https://doi.org/10.1007/978-3-030-70067-6
[14]	A. Candel, L. Conlon, Foliations, I, American Mathematical Society, Rhode Island, 2000.
[15]	P. Petersen, Riemannian Geometry, $3^{rd}$ edition, Springer, Cham, 2016.
[16]	M. Berger, A Panoramic View of Riemannian Geometry, Springer-Verlag, Heidelberg, 2003. https://doi.org/10.1007/978-3-642-18245-7

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)