[Paper Review] Scalar Curvature of Manifolds with Boundaries: Natural Questions and Artificial Constructions

Misha Gromov|arXiv (Cornell University)|Nov 10, 2018

Geometric Analysis and Curvature Flows1 references20 citations

TL;DR

This paper investigates geometric constraints on Riemannian manifolds with boundary under scalar curvature bounds, using spin geometry and Dirac operator techniques to derive sharp inequalities linking boundary mean curvature, scalar curvature, and the hyperspherical radius of the boundary. The key result establishes that for spin manifolds with non-negative scalar curvature, the infimum of the boundary mean curvature is bounded above by $\frac{n-1}{\mathrm{Rad}_{S^{n-1}}(Y)}$, with equality for Euclidean balls, and extends to negative scalar curvature via product constructions.

ABSTRACT

We present several problems and results relating the scalar curvatures of manifolds with mean curvatures of their boundaries

Motivation & Objective

• To determine necessary and sufficient geometric conditions on a closed Riemannian manifold $Y$ with a function $M$ for it to bound a compact manifold $X$ with $\mathrm{Sc}(X) \geq \sigma$ and mean curvature $M$ on $\partial X$.
• To investigate whether large mean curvature on $\partial X$ rules out fillings with non-negative scalar curvature, especially in spin manifolds.
• To establish lower bounds on the volume of such fillings $X$ in terms of the geometry of $\partial X$ and the scalar curvature bound $\sigma$, particularly for $\sigma = 0$.
• To explore analogies between mean curvature bounds and dihedral angle constraints in manifolds with corners, motivated by conjectures on extremal geometry.

Proposed method

• Uses the double $X \cup_Y X$ of a spin manifold $X$ with boundary $Y$ to apply Goette-Semmelmann's extremality theorem for metrics with positive curvature operator.
• Maps the double to a sphere $S^n$ equipped with a radial metric of positive curvature operator, leveraging spectral theory of the Dirac operator.
• Applies the inequality $[\mathrm{mean}]_{\mathrm{Sc} \geq 0}$ to $X \times S^m(R)$ with $R = \sqrt{m(m-1)/\sigma}$ to derive the $\mathrm{Sc} < 0$ case.
• Introduces the hyperspherical radius $\mathrm{Rad}_{S^{n-1}}(Y)$ as a measure of the size of $Y$, defined via degree-nonzero, distance-decreasing maps to spheres.
• Considers extremal cases such as Euclidean balls and their rigidity under scalar curvature and mean curvature constraints.
• Proposes conjectures on volume bounds and stability of manifolds with scalar curvature approaching that of the sphere, using intrinsic flat convergence and degree theory.

Experimental results

Research questions

• RQ1 Does sufficiently large mean curvature on the boundary $Y$ of a compact manifold $X$ rule out fillings with $\mathrm{Sc}(X) \geq 0$?
• RQ2 Is there a universal lower bound on the volume of a filling manifold $X$ with $\mathrm{Sc}(X) \geq \sigma$, in terms of the boundary geometry and mean curvature?
• RQ3 Can the dihedral angles of manifolds with corners be bounded from above in terms of scalar curvature and boundary geometry, especially for $\mathrm{Sc}(X) \geq 0$?
• RQ4 Does the rigidity of Euclidean space hold under scalar curvature non-negativity and asymptotic flatness, and how does it relate to Witten’s positive mass theorem?
• RQ5 Can one establish a volume comparison for manifolds with $\mathrm{Sc}(X) \geq -\varepsilon$, $\mathrm{Rad}_{S^{n-1}}(\partial X) \geq 1$, and $\mathrm{mean.curv}(\partial X) \geq n-1$?

Key findings

• For a compact, orientable spin manifold $X$ of dimension $n$ with $\mathrm{Sc}(X) \geq -\sigma$, the infimum of the mean curvature on $\partial X$ satisfies $\inf_{y \in Y} \mathrm{mean.curv}(Y,y) \leq \max\left(\frac{n+m-1}{\mathrm{Rad}_{S^{n-1}}(Y)}, \sqrt{\frac{\sigma}{m(m-1)}}\right)$ for all $m \geq 2$, with equality for Euclidean balls.
• In the case $\mathrm{Sc}(X) \geq 0$, the bound simplifies to $\inf_{y \in Y} \mathrm{mean.curv}(Y,y) \leq \frac{n-1}{\mathrm{Rad}_{S^{n-1}}(Y)}$, which is sharp and achieved by balls in $\mathbb{R}^n$.
• The inequality $[\mathrm{mean}]_{\mathrm{Sc} \geq 0}$ is proven via a double construction and mapping to a sphere with positive curvature operator, relying on Goette-Semmelmann’s theorem.
• The rigidity of Euclidean space is suggested: if $X^+$ is isometric to $\mathbb{R}^n$ outside a compact set and $\mathrm{Sc}(X^+) \geq 0$, then $X^+$ is isometric to $\mathbb{R}^n$, though the proof does not fully establish this due to smoothing errors.
• A conjecture is proposed that for $\mathrm{Sc}(X) \geq \sigma_n$, any smooth area-decreasing map from $X$ to $S^n$ that collapses a collar of $\partial X$ to a point must have degree zero.
• The spherical stability problem is posed: if $\mathrm{Sc}(X_i) \to n(n-1)$ and $\mathrm{Rad}_{S^n}(X_i) \to 1$, then $X_i$ should converge in intrinsic flat topology to $S^n$, possibly with thin bridges attached.