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Junrong Yan

Publications and source records attributed to Junrong Yan.

14 recordsLinked to original sources
arXiv2026

Large-Scale Regularity Meets the Weyl Law on Ricci Shrinkers

We establish a large-scale regularity property for complete gradient Ricci shrinkers and apply it to the spectral asymptotics of the weighted Laplacian. Although uniform bounded geometry is not known for general Ricci shrinkers, we prove that, inside large geodesic balls, the region where the curvature radius degenerates at the reciprocal scale occupies an asymptotically negligible proportion of the volume. The proof uses the Ricci flow induced by the shrinker, together with the compactness and regularity theories developed by Bamler and Li-Wang. As an application, we prove that the weighted Laplacian, or equivalently its conjugate Schr\"odinger operator, on every complete gradient Ricci shrinker satisfies the classical Weyl law.

math.DG↗
arXiv2026

Renormalizations in holomorphic field theories on K\"ahler manifolds

The divergence of Feynman graph integrals is one of the central issues in the study of perturbative quantum field theories. A rigorous formulation of these integrals usually requires renormalization. In this paper, we prove that the Feynman graph integrals arising from holomorphic field theories on closed real-analytic K\"ahler manifolds are convergent with respect to heat-kernel renormalization, Cauchy principal value renormalization, and zeta-function renormalization. Moreover, these three renormalization procedures produce the same value. Our proof is based on the theory of wonderful compactifications in algebraic geometry, which provides a geometric understanding of these integrals. As a consequence, we establish a gauge anomaly formula for these graph integrals.

math-ph↗
arXiv2026

The classical Weyl law for Schr\"odinger operators on complete Riemannian manifolds

We establish a criterion for the validity of the classical (non-semiclassical) Weyl law for Schr\"odinger operators $ H=\Delta+V $ on complete Riemannian manifolds. In contrast to existing results, our approach does not rely on standard geometric assumptions such as bounded geometry, nor on analytic assumptions such as the doubling condition on the potential. Instead, we identify a geometric-analytic invariant that encodes the precise balance between the geometry of the manifold, the growth of $V$, and the oscillation scale of $V$. This intrinsic quantity, denoted $c_{\delta}(\lambda)$ admits effective quantitative estimates. We prove that the Weyl asymptotic holds provided $\lim_{\lambda\to\infty} c_\delta(\lambda)=0 .$ The sharpness of this criterion is demonstrated through explicit examples showing that the Weyl law can fail when the criterion is violated.

math.DG↗
arXiv2025

Feynman Graph Integrals on K\"ahler Manifolds

In this paper, we establish the convergence of Feynman graph integrals on closed real-analytic K\"ahler manifolds and uncover the structural mechanism underlying this convergence. The key insight is that, using Getzler's rescaling technique, the graph integrands extend canonically to the Fulton-MacPherson compactification of configuration spaces as forms with divisorial-type singularities. This allows the Feynman graph integrals to be rigorously defined as Cauchy principal value integrals. As an application, these integrals provide a mathematically rigorous construction of the higher-genus B-model invariants on Calabi-Yau threefolds in the sense of Bershadsky-Cecotti-Ooguri-Vafa (BCOV).

math-ph↗
arXiv2025

A gluing formula for the $Z_2$-valued index of odd symmetric operators

We investigate Dirac-type operator $D$ on involutive manifolds with boundary with symmetry, which forces the index of $D$ to vanish. We study the secondary $Z_2$-valued index of elliptic boundary value problems for such operators. We prove a $Z_2$-valued analog of the splitting theorem: the $Z_2$-valued index of an operator on a closed manifold $M$ equals the $Z_2$-valued index of a boundary value problem on a manifold obtained by cutting $M$ along a hypersurface $N$. When $N$ divides $M$ into two disjoint submanifolds $M_1$ and $M_2$, the $Z_2$-valued index on $M$ is equal to the mod 2 reduction of the usual $Z$-valued index of the Atiyah-Patodi-Singer boundary value problem on $M_1$. This leads to a cohomological formula for the $Z_2$-valued index.

math.DG↗
arXiv2025

Weyl Law for Schr\"odinger Operators on Noncompact Manifolds, Heat Kernel, and Karamata-Hardy-Littlewood Theorem

Building on our earlier work on heat kernel asymptotics for Schr\"odinger-type operators on noncompact manifolds, we establish both the classical and semiclassical Weyl laws for Schr\"odinger operators of the form $\Delta+V$ and $\hbar^2\Delta+V$ on complete noncompact manifolds. While the semiclassical law can be approached via localization, the classical Weyl law has remained widely expected but unproven in this generality. We impose a mild bounded integral oscillation condition on $ V $ in addition to the assumptions that $V$ diverges at infinity and satisfies a doubling condition. In this setting, our oscillation condition is sharp and strictly weaker than all previously known assumptions, even in the Euclidean case. A central novelty of our approach is an extended Karamata-Hardy-Littlewood Tauberian theorem, adapted to accommodate non-regularly varying spectral asymptotics in noncompact settings, together with its semiclassical analogue. These Tauberian tools allow us to derive both versions of Weyl's law within a unified framework.

math.DG↗
arXiv2024

Generalized Morse Functions, Excision and Higher Torsions

Comparing invariants from both topological and geometric perspectives is a key focus in index theorem. This paper compares higher analytic and topological torsions and establishes a version of the higher Cheeger-M\"uller/Bismut-Zhang theorem. In fact, Bismut-Goette achieved this comparison assuming the existence of fiberwise Morse functions satisfying the fiberwise Thom-Smale transversality condition (TS condition). To fully generalize the theorem, we should remove this assumption. Notably, unlike fiberwise Morse functions, fiberwise generalized Morse functions (GMFs) always exist, we extend Bismut-Goette's setup by considering a fibration $ M \to S $ with a unitarily flat complex bundle $ F \to M $ and a fiberwise GMF $ f $, while retaining the TS condition. Compared to Bismut-Goette's work, handling birth-death points for a generalized Morse function poses a key difficulty. To deal with this, first, by the work of the author M.P., joint with Zhang and Zhu, we focus on a relative version of the theorem. Here, analytic and topological torsions are normalized by subtracting their corresponding torsions for trivial bundles. Next, using new techniques from by the author J.Y., we excise a small neighborhood around the locus where $f$ has birth-death points. This reduces the problem to Bismut-Goette's settings (or its version with boundaries) via a Witten-type deformation. However, new difficulties arise from very singular critical points during this deformation. To deal with these, we extend methods from Bismut-Lebeau, using Agmon estimates for noncompact manifolds developed by Dai and J.Y.

math.DG↗
arXiv2023

A first eigenvalue estimate for embedded hypersurfaces in positive Ricci curvature manifolds

Let $\Sigma$ be a closed, embedded, oriented hypersurface in a closed oriented Riemannian manifold $N$. Under a lower bound on the Ricci curvature and an upper bound on the sectional curvature of $N$, we establish a lower bound for the first nonzero eigenvalue of the Laplacian on $\Sigma$. The estimate depends on the ambient curvature bounds, the normal injectivity radius, and the geometry of $\Sigma$ through its mean curvature and second fundamental form. This result extends the classical eigenvalue estimate of Choi and Wang [J. Diff. Geom. \textbf{18} (1983), 559--562.] to the non-minimal case.

math.DG↗
arXiv2023

Witten deformation for non-Morse functions and gluing formula for analytic torsions

This paper concentrates on analyzing Witten deformation for a family of non-Morse functions parameterized by $T\in \mathbb{R}_+$, resulting in a novel, purely analytic proof of the gluing formula for analytic torsions in complete generality due to Br\"unning-Ma. Intriguingly, the gluing formula in this article could be reformulated as the Bismut-Zhang theorem for non-Morse functions, and from the perspective of Vishik's theory of moving boundary problems, the deformation parameter $T$ parameterize a family of boundary conditions. Our proof also makes use of a connection between small eigenvalues of Witten Laplacians and Mayer-Vietoris sequences. Finally, these new techniques could be extended to analytic torsion forms and play key roles in the study of the higher Cheeger-M\"uller/Bismut-Zhang theorem for nontrivial flat bundles.

math.DG↗
arXiv2022

Calabi-Yau/Landau-Ginzburg Correspondence for Weil-Peterson Metrics and $tt^*$ Structures

The aim of this paper is to rigorously establish the Calabi-Yau/Landau-Ginzburg (CY/LG) correspondence for the $tt^*$ geometry structure--a generalized version of variation of Hodge structures. Although it is well-known that there exists a map between Hodge structures on the LG and CY's sides that preserves the Hodge filtration and bilinear form, it remains unclear whether the real structures are also preserved. In our paper, we conduct a detailed analysis of two period integrals on the LG's side. Based on this analysis, we modify the real structure proposed by Cecotti on LG's side, and show that the aforementioned map is also preserved under the modified real structure. As a result, we establish full CY/LG correspondence for $tt^*$ structures.

math-ph↗
arXiv2020

Witten Deformation on Non-compact Manifold: Heat Kernel Expansion and Local Index Theorem

Asymptotic expansions of heat kernels and heat traces of Schrödinger operators on non-compact spaces are rarely explored, and even for cases as simple as $\mathbb{C}^n$ with (quasi-homogeneous) polynomials potentials, it's already very complicated. Motivated by path integral formulation of the heat kernel, we introduced a parabolic distance, which also appeared in Li-Yau's famous work on parabolic Harnack estimate. With the help of the parabolic distance, we derive a pointwise asymptotic expansion of the heat kernel for the Witten Laplacian with strong remainder estimate. When the deformation parameter of Witten deformation and time parameter are coupled, we derive an asymptotic expansion of trace of heat kernel for small-time $t$, and obtain a local index theorem. This is the second of our papers in understanding Landau-Ginzburg B-models on nontrivial spaces, and in subsequent work, we will develop the Ray-Singer torsion for Witten deformation in the non-compact setting.

math.DG↗
arXiv2020

Witten deformation for noncompact manifolds with bounded geometry

Motivated by the Landau-Ginzburg model, we study the Witten deformation on a noncompact manifold with bounded geometry, together with some tameness condition on the growth of the Morse function $f$ near infinity. We prove that the cohomology of the Witten deformation $d_{Tf}$ acting on the complex of smooth $L^2$ forms is isomorphic to the cohomology of Thom-Smale complex of $f$ as well as the relative cohomology of a certain pair $(M, U)$ for sufficiently large $T$. We establish an Agmon estimate for eigenforms of the Witten Laplacian which plays an essential role in identifying these cohomologies via Witten's instanton complex, defined in terms of eigenspaces of the Witten Laplacian for small eigenvalues. As an application we obtain the strong Morse inequalities in this setting.

math.DG↗