Rigidity and flexibility under spectral Ricci lower bounds and mean-convex boundary

Technical Summary: Rigidity and Flexibility Under Spectral Ricci Lower Bounds and Mean-Convex Boundary

Problem Statement
This paper investigates the geometric and topological rigidity of Riemannian manifolds $(M^n, g)$ with non-empty boundary $\partial M$, subject to spectral lower bounds on the Ricci curvature and mean-convex boundary conditions. The central inquiry addresses a boundary analogue of "Question 1" posed in the introduction: determining the sharp threshold $\gamma(n)$ such that if a compact manifold admits a metric satisfying the spectral condition $\lambda_1(-\gamma \Delta + \text{Ric}) \ge 0$ (and potentially a boundary condition), it must also admit a metric with strictly positive sectional curvature and strictly mean-convex boundary.

The spectral condition is defined via the existence of a positive function $u \in C^{2,\alpha}(M)$ such that $-\gamma \Delta u + u \text{Ric} \ge 0$ on $M$ and $H_{\partial M} + \gamma u^{-1}\langle \nabla u, \nu \rangle \ge 0$ on $\partial M$, where $H_{\partial M}$ is the mean curvature with respect to the outward unit normal $\nu$. The authors seek to extend classical rigidity theorems (specifically Kasue's splitting theorem for $\text{Ric} \ge 0$) to this spectral setting and determine the optimal range of $\gamma$ for which such rigidity or topological constraints hold.

Methodology
The authors employ a variational approach centered on weighted geodesics. Unlike recent works utilizing $\mu$-bubbles (which involve higher-dimensional minimal surfaces), this paper utilizes one-dimensional minimizers of the functional:
$$E(\beta) := \int_\beta u^\gamma d\ell_g$$
where $d\ell_g$ is the Riemannian length element. This choice allows the authors to avoid regularity issues associated with higher-dimensional minimal hypersurfaces in non-compact settings.

The core technical machinery involves:

1. First and Second Variation Formulas: Detailed computations are performed for the functional $E(\beta)$ along curves connecting disjoint boundary components. These yield stability inequalities involving the Ricci curvature, the Hessian of $u$, and boundary mean curvature terms.
2. Stability Inequalities (Lemma 1 & 2): By testing the second variation with specific cut-off functions and utilizing a weighted frame, the authors derive integral inequalities that constrain the geometry of minimizing curves.
3. Geodesic Capturing and Compactness (Lemma 3 & Proposition 4): A critical technical contribution is the proof that minimizing sequences for $E(\beta)$ connecting a compact boundary component to the rest of the boundary cannot "escape to infinity" in the non-compact case, provided $\gamma$ lies within a specific sharp range. This is achieved by constructing a contradiction using cut-off functions and the stability inequality, ensuring the existence of global minimizers.
4. Topological Arguments: The paper combines the geometric rigidity results with a deep theorem by Lawson and Michelsohn regarding the existence of metrics with positive sectional curvature.

Key Contributions and Results

1. Sharp Spectral Rigidity (Theorem 1)
The authors prove a sharp extension of Kasue's rigidity theorem. If $(M^n, g)$ is a complete manifold with disconnected boundary (at least one component compact) satisfying the spectral conditions with:

• $0 \le \gamma < 4$ for $n=2$,
• $0 \le \gamma < \frac{n-1}{n-2}$ for $n \ge 3$,
  then the manifold splits isometrically as a product $[0, L] \times \Sigma$, where $\Sigma$ is a compact manifold with $\text{Ric}_\Sigma \ge 0$.
• Sharpness: The authors provide counterexamples (Propositions 2 and 3) demonstrating that if $\gamma$ exceeds these thresholds, the splitting conclusion fails. Specifically, for $\gamma = \frac{n-1}{n-2}$, the manifold may fail to split, and for $\gamma > \frac{n-1}{n-2}$, the topological conclusions fail.

2. Topological Rigidity (Theorem 2)
The paper establishes that under the same spectral conditions (with the threshold extended to include equality for $n \ge 3$, i.e., $0 \le \gamma \le \frac{n-1}{n-2}$), if at least one inequality is strict, the relative fundamental group vanishes: $\pi_1(M, \partial M) = 0$. This implies the boundary is connected and the inclusion $\partial M \hookrightarrow M$ induces a surjection on fundamental groups.

• Significance: This generalizes known results for $\text{Ric} \ge 0$ to the spectral setting. The authors note that the compactness of the manifold itself is not required; only one boundary component must be compact.

3. Existence of Positive Curvature Metrics (Theorem 3)
Combining Theorem 2 with the Lawson–Michelsohn theorem, the authors prove that for $n \neq 4$, any compact manifold with boundary satisfying the spectral conditions (with at least one strict inequality) admits a metric with positive sectional curvature and strictly mean-convex boundary.

• Sharpness: The range of $\gamma$ is shown to be sharp via Proposition 1, which constructs manifolds with $\lambda_1(-\gamma \Delta + \text{Ric}) > 0$ that do not admit such metrics when $\gamma > \frac{n-1}{n-2}$.
• Dimensional Constraint: The result holds for $n \neq 4$. The authors explicitly state that the case $n=4$ remains open due to the unresolved status of a specific proposition in the Lawson–Michelsohn work regarding fake 4-cells.

4. Corollary 1 (Unified Rigidity)
This result synthesizes the previous theorems, stating that for the sharp range of $\gamma$, a manifold satisfying the spectral conditions must either split isometrically (if the boundary is disconnected) or have a trivial relative fundamental group (if the boundary is connected).

Significance and Claims
The paper claims to settle the sharp boundary analogue of Question 1 for dimensions $n \neq 4$. The authors emphasize that their use of weighted length minimizers (1-dimensional objects) rather than $\mu$-bubbles is a distinct methodological advantage, particularly in handling non-compact manifolds and avoiding higher-dimensional regularity issues.

The authors are modest regarding the scope of their work:

• They acknowledge that Question 1 remains widely open for $n \ge 5$ in the general compact case without boundary assumptions.
• They note that the compactness assumption in Theorem 1 (Item 2) might be removable for $n=2,3$, but this is left for future work.
• They explicitly state that the $n=4$ case is an open problem due to external constraints in the Lawson–Michelsohn theorem.
• They do not propose new applications or experimental proposals but frame their work as a theoretical advancement in geometric analysis, specifically connecting spectral geometry to topological rigidity and curvature bounds.

The paper concludes by providing Appendix results (Proposition 5 and Remark 6) that offer spectral generalizations of inradius and Bonnet–Myers type diameter estimates, showing slight improvements in the oscillation dependence of the bounds compared to previous literature.