The Vanishing of the Fundamental Gap of Convex Domains in $$\mathbb {H}^n$$

Integral Ricci curvature and the mass gap of Dirichlet Laplacians on domains

“…\end{equation*}$$This has been generalized to convex domains in $$\mathbb {S}^n$$ in [8, 13, 23], showing the same gap estimate. On the other hand, it was proven in [3] that given any diameter, there are convex domains $$\Omega \subset \mathbb {H}^n$$ with an arbitrarily small fundamental gap.…”

“…This is generalized to compact manifolds in [15], showing that the gap is bounded from below in terms of uniform lower bounds on the Ricci curvature, the diameter, sectional curvature near the boundary, and the interior rolling R ‐ball condition. The construction in [3] shows that the rolling R ‐ball condition is necessary there and for general manifolds, the rolling R ‐ball condition is more suitable than the convexity condition in some sense.…”

On the critical points of semi-stable solutions on convex domains of Riemannian surfaces

“…Here the situation is considerably different, Indeed, unlike the results in S 2 or R 2 , there exist geodesically convex domains for which the first Dirichlet eigenfunction has non-convex level sets, see [21]. Using similar ideas, in [22] the author constructs an example of a convex domain such that the corresponding first Dirichlet eigenfunction has two distinct maxima. Hence the convexity of the domain is not enough to guarantee the uniqueness of the critical point of the eigenfunction in the negatively curved case.…”

The fundamental gap of a kind of sub-elliptic operator

“…In particular, Laplacian fundamental gap problem [39,6,18,12] has been extended to different complex spaces and the corresponding detailed characterization is given. The lower bound of eigenvalue gap of vibrating string is also discussed in [22,8,37].…”