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Let $M^n$ be an $n$-dimensional smooth compact Riemannian manifold with boundary. Assume that the sectional curvature of $M$ is at least $\kappa$, the diameter is at most $D$, and the second fundamental form of the boundary is at least $\lambda$.

Question. Does there exist a lower bound on the sectional curvature of the boundary $\partial M$ in terms of $n,\kappa,\lambda$ and (possibly) $D$?

Even partial results under extra assumptions might be helpful.

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  • $\begingroup$ What do you mean by "second fundamental form of the boundary is at least λ"? Do you want to compare with the metric on the boundary? And have you already tried to use the classical embedding formulas for hypersurfaces? $\endgroup$
    – Sebastian Goette
    Commented Mar 1, 2017 at 19:17
  • $\begingroup$ @SebastianGoette: I mean that on the boundary the second fundamental form is at least $\lambda g$, where $g$ is the original metric. $\endgroup$
    – asv
    Commented Mar 1, 2017 at 19:30

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I assume that convex surfaces have nonnegative second fundamental forms.

If $\lambda\geqslant 0$ then the answer is yes, and it follows from the Gauss formula.

On the other hand, if $\lambda<0$, then it implies no bound on sectional curvature. This can be seen already for surfaces in Euclidean space.

However, for $\lambda<0$, the Gauss formula implies the following inequality, which is helpful sometimes: $$\mathrm{sec}_{\partial M}\geqslant \kappa + \lambda\cdot (H-n\cdot\lambda);$$ here $H$ denotes the mean curvature of the boundary. Again, we assume that $H\geqslant0$ for convex hypersurfaces.

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