New spectral Bishop-Gromov and Bonnet-Myers theorems and applications to isoperimetry

Technical Summary: New Spectral Bishop–Gromov and Bonnet–Myers Theorems and Applications to Isoperimetry

Problem Statement
The paper addresses the generalization of classical comparison theorems in Riemannian geometry—specifically the Bishop–Gromov volume comparison theorem and the Bonnet–Myers diameter theorem—to a spectral setting. The classical theorems rely on pointwise lower bounds on the Ricci curvature ($\text{Ric} \geq (n-1)\lambda$). This work investigates whether similar volume and diameter bounds hold when the curvature condition is replaced by a spectral condition involving the operator $-\gamma\Delta + \text{Ric}$, where $\gamma \geq 0$.

The central problem is to determine the sharp range of the parameter $\gamma$ for which these spectral conditions imply:

1. Finiteness of the fundamental group $\pi_1(M)$.
2. Upper bounds on the diameter of the universal cover $\tilde{M}$.
3. Upper bounds on the volume of $M$ (or $\tilde{M}$).
4. Sharp isoperimetric control at infinity for complete manifolds with nonnegative Ricci curvature and spectral bi-Ricci curvature bounds.

Methodology
The authors employ a combination of geometric measure theory, spectral analysis, and variational methods. The core technical innovations include:

1. Unequally Weighted Isoperimetric Profiles:
   Unlike previous spectral generalizations (e.g., in dimension 3) that used equally weighted profiles, the authors introduce a profile with unequal weights for the area and volume terms. For a positive function $u$ satisfying the spectral condition, they define:
   $$I(v) := \inf \left\{ \int_{\partial^* E} u^\gamma : E \subset \subset M, \int_E u^{\frac{2\gamma}{n-1}} = v \right\}$$
   This specific weighting allows the derivation of a sharp differential inequality for the isoperimetric profile:
   $$I''I \leq -\frac{(I')^2}{n-1} - (n-1)\lambda$$
   This inequality holds in the viscosity sense and matches the sharp inequality known for the constant curvature case, enabling a sharp volume bound via ODE comparison.

2. Unequally Warped $\mu$-Bubbles:
   To establish diameter bounds, the authors utilize the $\mu$-bubble technique (originally due to Gromov). They define a functional with an unequally warped potential:
   $$E(\Omega) = \int_{\partial^* \Omega} u^\gamma - \int_{\Omega} h u^{\frac{2\gamma}{n-1}}$$
   By carefully choosing the warping function $h$ to satisfy a specific differential inequality involving $\nabla h$, $u$, and $\gamma$, they derive a contradiction if the diameter is too large. This approach extends the diameter bound to the full range of $\gamma$ where the spectral condition is meaningful.

3. Stability and Regularity:
   The proofs involve analyzing the second variation of the energy functional. A significant technical hurdle is the potential formation of singularities in minimizing hypersurfaces for dimensions $n \geq 8$. The authors adapt arguments from the literature (specifically [7, 11]) to show that stability inequalities hold for variations supported away from the singular set, thereby extending their results to all dimensions $n \geq 3$.

Key Contributions and Results

1. Spectral Bishop–Gromov and Bonnet–Myers Theorems (Theorem 1 & Corollary 1)
The paper establishes a sharp generalization of the classical theorems. Let $(M, g)$ be a closed Riemannian manifold of dimension $n \geq 3$. If there exists a positive smooth function $u$ such that:
$$\gamma \Delta u \leq u \text{Ric} - (n-1)\lambda u$$
with $0 \leq \gamma \leq \frac{n-1}{n-2}$ and $\lambda > 0$, then:

• Diameter Bound: The diameter of the universal cover $\tilde{M}$ is bounded by:
  $$\text{diam}(\tilde{M}) \leq \frac{\pi}{\sqrt{\lambda}} \left( \frac{\max(u)}{\min(u)} \right)^{\frac{n-3}{n-1}\gamma}$$
  Consequently, $\pi_1(M)$ is finite.
• Volume Bound: The volume of the universal cover satisfies:
  $$\text{vol}(\tilde{M}) \leq \lambda^{-n/2} \text{vol}(S^n)$$
• Rigidity: If equality holds in the volume bound, then $u$ is constant, and $\tilde{M}$ is isometric to the round sphere of radius $\lambda^{-1/2}$.

The authors prove that the range $\gamma \leq \frac{n-1}{n-2}$ is sharp; for $\gamma > \frac{n-1}{n-2}$, counterexamples exist (e.g., on $S^1 \times S^{n-1}$) where the spectral condition holds but the volume and diameter bounds fail.

2. Solution to the Stable Bernstein Problem (Theorem 2)
As a direct application, the authors resolve the stable Bernstein problem in $\mathbb{R}^6$. They prove that any immersed, complete, connected, two-sided, stable minimal hypersurface $M^n \hookrightarrow \mathbb{R}^{n+1}$ for $2 \leq n \leq 5$ is a Euclidean hyperplane. This result relies on the spectral condition derived from the stability operator and the sharp range of $\gamma$ established in Theorem 1. The case $n=6$ remains open.

3. Isoperimetry at Infinity (Theorem 3)
For complete noncompact manifolds with $3 \leq n \leq 5$, $\text{Ric} \geq 0$, and a spectral bi-Ricci condition $\lambda_1(-\gamma\Delta + \text{biRic}) \geq n-2$ outside a compact set, the paper proves:

• Linear Volume Growth: The manifold has linear volume growth.
• Sharp Isoperimetric Profile: The isoperimetric profile $I_M(v)$ satisfies $I_M(v) \leq \text{vol}(S^{n-1})$ for all $v > 0$.
• Rigidity: Equality for some $v$ implies the manifold is isometric to a cylinder $S^{n-1} \times [0, \infty)$ outside a bounded set.

Significance and Claims
The authors claim that the primary novelty lies in the introduction of the unequally weighted isoperimetric profile and unequally warped $\mu$-bubbles. These tools allow for the recovery of sharp constants and the extension of results to the full range of $\gamma$ where the spectral condition is valid, overcoming limitations of previous methods that required stricter bounds on $\gamma$ (e.g., $\gamma < \frac{4}{n-1}$) or yielded non-sharp volume bounds.

The paper emphasizes that the range $\gamma \leq \frac{n-1}{n-2}$ is optimal for the finiteness of $\pi_1(M)$ and volume bounds, a fact demonstrated by explicit counterexamples for supercritical $\gamma$. Furthermore, the work unifies and extends previous results on the stable Bernstein problem and isoperimetric structures in manifolds with nonnegative Ricci curvature, providing a unified spectral framework for these geometric problems. The results are presented as applicable to all dimensions $n \geq 3$, with specific handling for the singular case $n \geq 8$.