Abstract
Let \((M,{\bar{g}})\) be a compact Riemannian manifold with minimal boundary such that the second fundamental form is nowhere vanishing on \(\partial M\). We show that for a generic Riemannian metric \({\bar{g}}\), the squared norm of the second fundamental form is a Morse function, i.e. all its critical points are non-degenerate. We show that the generality of this property holds when we restrict ourselves to the conformal class of the initial metric on M.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
References
Ahmedou, M., Ben Ayed, M.: Non simple blow ups for the Nirenberg problem on half spheres. Discrete Contin. Dyn. Syst. 42(12), 5967–6005 (2022)
Almaraz, S.: A compactness theorem for scalar-flat metrics on manifolds with boundary. Calc. Var. Partial Differ. Equ. 41(3–4), 341–386 (2011)
Ben Ayed, M., Ghoudi, R., Ould Bouh, K.: Existence of conformal metrics with prescribed scalar curvature on the four dimensional half sphere. Nonlinear Differ. Equ. Appl. 19, 629–662 (2012)
Chang, Y.: Constant mean curvature hypersurfaces with two principal curvatures in a sphere. Monatsh. Math. 158(1), 1–22 (2009)
Cruz-Blázquez, S., Malchiodi, A., Ruiz, D.: Conformal metrics with prescribed scalar and mean curvature. J. Reine Angew. Math. 789, 211–251 (2022)
Cruz-Blázquez, S., Pistoia, A., Vaira, G.: Clustering phenomena in low dimensions for a boundary Yamabe problem, preprint 2022. arxiv:2211.08219
Djadli, Z., Malchiodi, A., Ahmedou, M.O.: Prescribing scalar and boundary mean curvature on the three dimensional half sphere. J. Geom. Anal. 13(2), 255–289 (2003)
Escobar, J.F.: Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary. Ann. Math. 2(136), 1–50 (1992)
Escobar, J.F.: The Yamabe problem on manifolds with boundary. J. Differ. Geom. 35(1), 21–84 (1992)
Ghimenti, M., Micheletti, A.M., Pistoia, A.: Blow-up phenomena for linearly perturbed Yamabe problem on manifolds with umbilic boundary. J. Differ. Equ. 267(1), 587–618 (2019)
Ghimenti, M., Micheletti, A.M., Pistoia, A.: Linear perturbation of the Yamabe problem on manifolds with boundary. J. Geom. Anal. 28, 1315–1340 (2018)
Ghimenti, M., Micheletti, A.M.: Nondegeneracy of critical points of the mean curvature of the boundary for Riemannian manifolds. J. Fixed Point Theory Appl. 14, 71–78 (2013)
Henry, D.: Perturbation of the boundary in boundary-value problems of partial differential equations. London Mathematical Society Lecture Note Series, 318. Cambridge University Press, Cambridge, 2005
Lawson, H.B., Jr.: Complete minimal surfaces in \(S^3\). Ann. Math. 92(2), 335–374 (1970)
Lee, J.M.: Introduction to Riemannian Manifolds. Graduate Texts in Mathematics, vol. 176, 2nd edn. Springer International Publishing, Berlin (2018)
Li, Y.Y.: The Nirenberg problem in a domain with boundary. Topol. Methods Nonlinear Anal. 6(2), 309–329 (1995)
Pinkall, U., Sterling, I.: On the classification of constant mean curvature tori. Ann. Math. 130(2), 407–451 (1989)
Thom, R.: Quelques propriétés globales des variétés différentiables. Comment. Math. Helv. 28, 17–86 (1954)
Acknowledgements
We express our gratitude to the referee for the careful reading of this work and for their valuable observations on the class of metrics \({\mathscr {E}}^m\).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest
The authors declare that they have no conflict of interest and no relevant financial or non-financial interests to disclose.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
S. Cruz has been partially supported by INdAM—GNAMPA Project 2022 “Fenomeni di blow-up per equazioni non lineari”, E55F22000270001S and by PRIN 2017JPCAPN. A. Pistoia has been partially supported by GNAMPA, Italy as part of INdAM.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Cruz-Blázquez, S., Pistoia, A. Non-degeneracy of Critical Points of the Squared Norm of the Second Fundamental Form on Manifolds with Minimal Boundary. J Geom Anal 33, 332 (2023). https://doi.org/10.1007/s12220-023-01395-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12220-023-01395-7
Keywords
- Prescribed curvature problem
- Conformal metric
- Morse function
- Second fundamental form
- Minimal boundary
- Transversality theorem