Local rigidity, bifurcation, and stability of Hf-hypersurfaces in weighted Killing warped products.

• Autores: Marco A. L. Velásquez, Henrique F. de Lima, Andre F. A. Ramalho
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 65, Nº 1, 2021, págs. 363-388
• Idioma: inglés
• DOI: 10.5565/publicacionsmatematiques.v65i1.384462
• Enlaces
  • Texto completo
• Resumen

  • In a weighted Killing warped product Mn f ×ρR with warping metric h , iM+ρ2 dt, where the warping function ρ is a real positive function defined on Mn and the weighted function f does not depend on the parameter t ∈ R, we use equivariant bifurcation theory in order to establish sufficient conditions that allow us to guarantee the existence of bifurcation instants, or the local rigidity for a family of open sets {Ωγ}γ∈I whose boundaries ∂Ωγ are hypersurfaces with constant weightedmean curvature. For this, we analyze the number of negative eigenvalues of a certain Schr¨odinger operator and study its evolution. Furthermore, we obtain a characterization of a stable closed hypersurface x: Σn ,→ Mn f ×ρ R with constant weighted mean curvature in terms of the first eigenvalue of the f-Laplacian of Σn.

• Referencias bibliográficas
  • L. J. Alías, M. Dajczer, and J. Ripoll, A Bernstein-type theorem for Riemannian manifolds with a Killing field, Ann. Global Anal. Geom. 31(4)...
  • L. J. Alías and P. Piccione, Bifurcation of constant mean curvature tori in Euclidean spheres, J. Geom. Anal. 23(2) (2013), 677–708. DOI:...
  • D. Bakry and M. Emery , Diffusions hypercontractives, in: “S´eminaire de Probabilit´es XIX 1983/84”, Lecture Notes in Math. 1123, Springer,...
  • L. Barbosa and P. Berard ´ , Eigenvalue and “twisted”’ eigenvalue problems, applications to CMC surfaces, J. Math. Pures Appl. (9) 79(5) (2000),...
  • J. L. Barbosa and M. do Carmo, Stability of hypersurfaces with constant mean curvature, Math. Z. 185(3) (1984), 339–353. DOI: 10.1007/BF01215045.
  • J. L. Barbosa, M. do Carmo, and J. Eschenburg, Stability of hypersurfaces of constant mean curvature in Riemannian manifolds, Math. Z. 197(1)...
  • M. Batista, M. P. Cavalcante, and J. Pyo, Some isoperimetric inequalities and eigenvalue estimates in weighted manifolds, J. Math. Anal. Appl....
  • S. Bernstein, Sur les surfaces d´efinies au moyen de leur courbure moyenne ou totale, Ann. Sci. Ecole Norm. Sup. (3) ´ 27 (1910), 233–256....
  • R. G. Bettiol, P. Piccione, and G. Siciliano, Equivariant bifurcation in geometric variational problems, in: “Analysis and Topology in Nonlinear...
  • R. G. Bettiol, P. Piccione, and G. Siciliano, On the equivariant implicit function theorem with low regularity and applications to geometric...
  • K. Castro and C. Rosales, Free boundary stable hypersurfaces in manifolds with density and rigidity results, J. Geom. Phys. 79 (2014), 14–28....
  • A. W. Cunha, E. L. de Lima, H. F. de Lima, E. A. Lima, Jr., and A. A. Medeiros, Bernstein type properties of two-sided hypersurfaces immersed...
  • M. Dajczer and J. H. S. de Lira, Entire bounded constant mean curvature Killing graphs, J. Math. Pures Appl. (9) 103(1) (2015), 219–227. DOI:...
  • M. Dajczer and J. H. de Lira, Entire unbounded constant mean curvature Killing graphs, Bull. Braz. Math. Soc. (N.S.) 48(2) (2017), 187–198....
  • M. Dajczer, P. A. Hinojosa, and J. H. de Lira, Killing graphs with prescribed mean curvature, Calc. Var. Partial Differential Equations 33(2)...
  • L. L. De Lima, J. H. De Lira, and P. Piccione, Bifurcation of Clifford tori in Berger 3-spheres, Q. J. Math. 65(4) (2014), 1345–1362. DOI:...
  • H. F. de Lima, E. Lima, Jr., A. Medeiros, and M. S. Santos, Liouville type results for two-sided hypersurfaces in weighted Killing warped...
  • J. M. Espinar, Gradient Schr¨odinger operators, manifolds with density and applications, J. Math. Anal. Appl. 455(2) (2017), 1505–1528. DOI:...
  • F. Fang, X.-D. Li, and Z. Zhang, Two generalizations of Cheeger–Gromoll splitting theorem via Bakry–Emery Ricci curvature, Ann. Inst. Fourier...
  • S. C. Garc´ıa-Mart´ınez and J. Herrera, Rigidity and bifurcation results for CMC hypersurfaces in warped product spaces, J. Geom. Anal. 26(2)...
  • M. Gromov, Isoperimetry of waists and concentration of maps, Geom. Funct. Anal. 13(1) (2003), 178–215. DOI: 10.1007/s000390300004.
  • D. Impera and M. Rimoldi, Stability properties and topology at infinity of f-minimal hypersurfaces, Geom. Dedicata 178 (2015), 21–47. DOI:...
  • M. Koiso, B. Palmer, and P. Piccione, Bifurcation and symmetry breaking of nodoids with fixed boundary, Adv. Calc. Var. 8(4) (2015), 337–370....
  • M. Koiso, B. Palmer, and P. Piccione, Stability and bifurcation for surfaces with constant mean curvature, J. Math. Soc. Japan 69(4) (2017),...
  • A. Lichnerowicz, Vari´et´es k¨ahleriennes `a premi`ere classe de Chern non negative et vari´et´es riemanniennes `a courbure de...
  • B. O’Neill, “Semi-Riemannian Geometry With Applications to Relativity”, Pure and Applied Mathematics 103, Academic Press, Inc. [Harcourt Brace...
  • G. Wei and W. Wylie, Comparison geometry for the Bakry–Emery Ricci tensor, J. Differential Geom. 83(2) (2009), 377–405. DOI: 10.4310/jdg/1261495...
  • L. Zeng, Eigenvalues of the drifting Laplacian on complete noncompact Riemannian manifolds, Nonlinear Anal. 141 (2016), 1–15. DOI: 10.1016/j.na....