Minimal hypersurfaces with small first eigenvalue in manifolds of positive Ricci curvature

@article{Zhu2015MinimalHW,
  title={Minimal hypersurfaces with small first eigenvalue in manifolds of positive Ricci curvature},
  author={Jonathan J. Zhu},
  journal={arXiv: Differential Geometry},
  year={2015}
}
  • Jonathan J. Zhu
  • Published 1 December 2015
  • Mathematics
  • arXiv: Differential Geometry
In this paper we exhibit deformations of the hemisphere $S^{n+1}_+$, $n\geq 2$, for which the ambient Ricci curvature lower bound $\text{Ric}\geq n $ and the minimality of the boundary are preserved, but the first Laplace eigenvalue of the boundary decreases. The existence of these metrics suggests that any resolution of Yau's conjecture on the first eigenvalue of minimal hypersurfaces in spheres would likely need to consider more geometric data than a Ricci curvature lower bound. 
Geometric Variational Problems for Mean Curvature
This thesis investigates variational problems related to the concept of mean curvature on submanifolds. Our primary focus is on the area functional, whose critical points are the minimal submanifoldsExpand
Index of Embedded Networks in the Sphere
  • Gaoming Wang
  • Mathematics
  • 2021
In this paper, we will compute the Morse index and nullity for the stationary embedded networks in spheres. The key theorem in the computation is that the index (and nullity) for the whole network isExpand

References

SHOWING 1-10 OF 27 REFERENCES
Isoparametric foliation and Yau conjecture on the first eigenvalue
A well known conjecture of Yau states that the first eigenvalue of every closed minimal hypersurface $M^n$ in the unit sphere $S^{n+1}(1)$ is just its dimension $n$. The present paper shows that YauExpand
Deformations of the hemisphere that increase scalar curvature
Consider a compact Riemannian manifold M of dimension n whose boundary ∂M is totally geodesic and is isometric to the standard sphere Sn−1. A natural conjecture of Min-Oo asserts that if the scalarExpand
Volume growth, eigenvalue and compactness for self-shrinkers
In this paper, we show an optimal volume growth for self-shrinkers, and estimate a lower bound of the first eigenvalue of $\mathcal{L}$ operator on self-shrinkers, inspired by the first eigenvalueExpand
The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature
In this paper we obtain a curvature estimate for embedded minimal surfaces in a three-dimensional manifold of positive Ricci curvature in terms of the geometry of the ambient manifold and the genusExpand
Splitting the Spectrum of a Riemannian Manifold
The eigenvalues of a Riemannian manifold have been calculated mostly only for spaces having a high degree of symmetry. In these cases the eigenvalues generally have large multiplicities. However, forExpand
TT-eigentensors for the Lichnerowicz Laplacian on some asymptotically hyperbolic manifolds with warped products metrics
Let (M =]0, ∞[×N, g) be an asymptotically hyperbolic manifold of dimension n +  1 ≥  3, equipped with a warped product metric. We show that there exist no TT L2-eigentensors with eigenvalue in theExpand
Laplacian eigenvalue functionals and metric deformations on compact manifolds
Abstract In this paper, we investigate critical points of the eigenvalues of the Laplace operator considered as functionals on the space of Riemannian metrics or a conformal class of metrics on aExpand
Rigidity Theorems for Compact Manifolds with Boundary and Positive Ricci Curvature
We prove some boundary rigidity results for the hemisphere under a lower bound for Ricci curvature. The main result can be viewed as the Ricci version of a conjecture of Min-Oo.
Mean Curvature Flow in a Ricci Flow Background
Following work of Ecker (Comm Anal Geom 15:1025–1061, 2007), we consider a weighted Gibbons-Hawking-York functional on a Riemannian manifold-with-boundary. We compute its variational properties andExpand
Spectra and symmetric eigentensors of the Lichnerowicz Laplacian on $P^n(\comp)$
We compute the eigenvalues with multiplicities of the Lichnerowicz Laplacian acting on the space of symmetric covariant tensor fields on the Euclidian sphere $S^n$. The spaces of symmetricExpand
...
1
2
3
...