Mixed Radial Volume Comparison under Spectral Ricci Bounds

Wed, 02 Sep 2026 00:00:00 +0000

Main results

This paper develops radial volume comparison under a spectral lower bound for Ricci curvature. The mixed radial construction separates the conformal geometry used to choose rays from the weights used to truncate and measure them.

Sharp comparison

A space-form model for mixed radial balls

Under the spectral inequality $\operatorname{Ric}_g-\alpha(\Delta_g u/u)g\geq(n-1)\kappa g$, the weighted volume of the $g$-arclength mixed radial domain satisfies a scale-invariant comparison with the corresponding space-form volume.

Polynomial growth

No pointwise bounds on the spectral weight

For an explicit range of radial and measure exponents, mixed radial balls have polynomial weighted-volume growth. The optimal exponent can attain the Euclidean value $n$.

Applications

Diameter, sharp volume, and stable Bernstein

The comparison gives radial proofs of the spectral Bonnet--Myers and sharp volume theorem of Antonelli--Xu, together with a short volume-growth proof that every complete two-sided stable minimal immersion $M^3\to\mathbb R^4$ is flat.

Proof architecture

The argument begins with an exact Riccati identity for the weighted radial Jacobian along geodesics of the conformal metric $h_\alpha=u^{2\alpha}g$. Taking the $(n-1)$-st root of this Jacobian converts the identity into a one-dimensional Sturm comparison and yields the space-form estimate. For general mixed radial balls, the logarithmic derivatives of the Jacobian and the spectral weight are instead treated as coupled variables; a quadratic estimate and a Bellman-type correction produce the polynomial-growth monotonicity formula. The positive-curvature case also gives an optimal diameter bound among smooth closed examples.

Spectral splitting theorem and ends of minimal hypersurfaces

Thu, 14 May 2026 15:06:46 +0000

Main results

The paper develops a geodesic-line approach to spectral Ricci curvature and uses it to control the topology at infinity of minimal hypersurfaces.

Spectral splitting

A weighted minimizing line recovers classical rigidity

If $\lambda_1(-\alpha\Delta+\operatorname{Ric})\geq0$ with $\alpha<4/(n-1)$ and the associated weighted metric admits a minimizing line, then $\operatorname{Ric}\geq0$ pointwise and the manifold splits off an $\mathbb R$ factor.

Ends at infinity

Infinitely many ends create a weighted line

Nonnegative spectral Ricci curvature outside a compact set, together with infinitely many ends, forces the existence of a weighted minimizing geodesic line outside a larger compact set.

Minimal hypersurfaces

Finite index implies finitely many ends

Every complete two-sided finite-index minimal hypersurface in an ambient manifold of dimension at most five with nonnegative biRic curvature has finitely many ends, without a properness assumption.

Proof strategy

The splitting proof adapts the Cheeger–Gromoll Busemann-function argument to the weighted length functional determined by a positive solution of the spectral equation. For the application, infinitely many ends are used to construct a weighted line at infinity; the positive solution is then perturbed to make the spectral inequality strict outside a compact set. The second variation of weighted length produces the contradiction and hence finiteness of ends.

Spectral Curvature | Gaoming Wang

Mixed Radial Volume Comparison under Spectral Ricci Bounds

Main results

A space-form model for mixed radial balls

No pointwise bounds on the spectral weight

Diameter, sharp volume, and stable Bernstein

Proof architecture

Spectral splitting theorem and ends of minimal hypersurfaces

Main results

A weighted minimizing line recovers classical rigidity

Infinitely many ends create a weighted line

Finite index implies finitely many ends

Proof strategy