Paper · Sep 8, 2026
The Weyl Law Meets Large-Scale Regularity on Ricci Shrinkers
As on arXiv
Abstract
We prove that the weighted Laplacian, or equivalently its conjugate Schrödinger operator, on every complete gradient Ricci shrinker satisfies the classical Weyl law. The main difficulty is that uniform bounded geometry is not known for general Ricci shrinkers. To overcome this, we establish a large-scale regularity property for complete gradient Ricci shrinkers and apply it to the spectral asymptotics of the weighted Laplacian. We prove that, inside large geodesic balls of radius , the region where the curvature radius is smaller than occupies an asymptotically negligible proportion of the volume. The proof uses the Ricci flow associated with the shrinker, together with the curvature-radius estimates and Sobolev inequalities of Li–Wang.
- Submitted
- Sep 8, 2026
- Last revised
- Sep 11, 2026 (v2)
- Subjects
- math.DG, math.SP
Comments: 29 pages. The proof of Lemma 4.4 has been greatly simplified. Comments are welcome!
Its neighbourhood
Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
-
Spectral bounds for -Laplace-type operators with applications to Betti numbers on gradient Ricci shrinkers
Topics Eigenvalues & spectrum, Ricci solitons · Shared terms “Ricci shrinkers”, “gradient Ricci”, “complete gradient”
-
The Ricci tensor of a gradient Ricci soliton with harmonic Weyl tensor
Topics Eigenvalues & spectrum, Ricci solitons · Shared terms “Weyl”, “gradient Ricci”
-
Eigenvalue Estimates for Schrödinger Operators on Ricci Shrinkers
Topics Eigenvalues & spectrum, Ricci solitons · Shared terms “Schrödinger operator”, “Ricci shrinkers”
-
The dimension of polynomial growth holomorphic functions and forms on gradient Kähler Ricci shrinkers
Topics Eigenvalues & spectrum, Ricci solitons · Shared terms “Ricci shrinkers”, “complete gradient”, “Laplacian”
-
Eigenvalue lower bounds and splitting for modified Ricci flow
Topics Eigenvalues & spectrum · Shared terms “Laplacian”
-
Hadamard-type variation formulae for the eigenvalues of a class of second-order elliptic operators and its applications
Topics Eigenvalues & spectrum
-
Eigenvalue estimates on shrinkers
Topics Eigenvalues & spectrum, Ricci solitons
-
The spectral rigidity of Ricci soliton and Einstein-type manifolds
Topics Eigenvalues & spectrum, Ricci solitons · Shared terms “spectral”, “Laplacian”
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 11B.1Ricci Solitons
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.
- 12B.2The Asymptotic Soliton and Three-Dimensional Shrinkers
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.
- 5A.5Uniformization and the Two-Dimensional Ricci Flow
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.