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Heat Flow in Semi-Flat Elliptic Fibrations

Xin Yu Liao

TL;DR

This work analyzes heat flow under SYZ-type collapse of Ricci-flat Kähler manifolds fibred by flat tori, showing that fiberwise-compressed heat semigroups converge to the base heat semigroup on precompact regions away from discriminant points. A central innovation is the conic renormalization at the discriminant, which defines a canonical limit functional K_B^{ren} by patching interior base heat-kernel pairings with the flat-cone model near D. The authors establish Mosco convergence of the Dirichlet forms for the interior regime and derive strong convergence of the compressed semigroups, then construct a conic parametrix to prove the existence and independence of the renormalized bilinear functional. Under exponentially small semi-flat errors, they obtain explicit exponential convergence rates and extend the framework to various boundary conditions, with a general geometric template for applying the renormalized limit in semi-flat elliptic fibrations. The results provide a rigorous, intrinsic mechanism to pass from interior collapse to a global, discriminant-inclusive heat-flow limit with precise quantitative control, offering a robust tool for understanding SYZ-type degenerations in mirror-symmetric geometries.

Abstract

Motivated by the SYZ picture for the collapsing of elliptic K3 surfaces, we study heat kernels under semi-flat collapse of Ricci-flat Kähler manifolds (X_t, g(t)) fibered by flat 2-tori over a surface B with a finite discriminant set D. On the regular locus B_reg = B \ D we assume an exponentially accurate semi-flat product approximation together with a uniform vertical spectral gap at the collapse scale. Using normalized lift and fiber-average maps, we show that for each fixed time tau > 0 the fiber-compressed heat operators converge strongly on L^2 to the heat semigroup of the base Laplacian. Equivalently, bilinear pairings of the total-space heat kernel against fiber-constant lifts of test functions converge on any precompact subset of B_reg. The main additional issue occurs at the discriminant: although X_t is smooth, the limiting base geometry becomes conic near D, so test functions meeting D cannot be treated by purely interior arguments. In wedge charts around D we introduce a canonical conic-renormalized bilinear functional obtained by patching the base heat-kernel pairing away from D with the model heat kernel of the corresponding flat cone. A conic parametrix shows that this functional is well-defined and independent of auxiliary cutoffs, and that the total-space bilinear pairings converge to it for arbitrary smooth compactly supported test functions and every fixed tau > 0. When the supports avoid D, the renormalization is trivial and the limit reduces to the usual base heat-kernel pairing. Under exponentially small geometric errors we obtain an explicit exponential convergence rate, uniform for tau in compact subintervals of (0, infinity). The results extend to Neumann and nonnegatively weighted Robin boundary conditions and globalize by exhaustion of the regular locus.

Heat Flow in Semi-Flat Elliptic Fibrations

TL;DR

This work analyzes heat flow under SYZ-type collapse of Ricci-flat Kähler manifolds fibred by flat tori, showing that fiberwise-compressed heat semigroups converge to the base heat semigroup on precompact regions away from discriminant points. A central innovation is the conic renormalization at the discriminant, which defines a canonical limit functional K_B^{ren} by patching interior base heat-kernel pairings with the flat-cone model near D. The authors establish Mosco convergence of the Dirichlet forms for the interior regime and derive strong convergence of the compressed semigroups, then construct a conic parametrix to prove the existence and independence of the renormalized bilinear functional. Under exponentially small semi-flat errors, they obtain explicit exponential convergence rates and extend the framework to various boundary conditions, with a general geometric template for applying the renormalized limit in semi-flat elliptic fibrations. The results provide a rigorous, intrinsic mechanism to pass from interior collapse to a global, discriminant-inclusive heat-flow limit with precise quantitative control, offering a robust tool for understanding SYZ-type degenerations in mirror-symmetric geometries.

Abstract

Motivated by the SYZ picture for the collapsing of elliptic K3 surfaces, we study heat kernels under semi-flat collapse of Ricci-flat Kähler manifolds (X_t, g(t)) fibered by flat 2-tori over a surface B with a finite discriminant set D. On the regular locus B_reg = B \ D we assume an exponentially accurate semi-flat product approximation together with a uniform vertical spectral gap at the collapse scale. Using normalized lift and fiber-average maps, we show that for each fixed time tau > 0 the fiber-compressed heat operators converge strongly on L^2 to the heat semigroup of the base Laplacian. Equivalently, bilinear pairings of the total-space heat kernel against fiber-constant lifts of test functions converge on any precompact subset of B_reg. The main additional issue occurs at the discriminant: although X_t is smooth, the limiting base geometry becomes conic near D, so test functions meeting D cannot be treated by purely interior arguments. In wedge charts around D we introduce a canonical conic-renormalized bilinear functional obtained by patching the base heat-kernel pairing away from D with the model heat kernel of the corresponding flat cone. A conic parametrix shows that this functional is well-defined and independent of auxiliary cutoffs, and that the total-space bilinear pairings converge to it for arbitrary smooth compactly supported test functions and every fixed tau > 0. When the supports avoid D, the renormalization is trivial and the limit reduces to the usual base heat-kernel pairing. Under exponentially small geometric errors we obtain an explicit exponential convergence rate, uniform for tau in compact subintervals of (0, infinity). The results extend to Neumann and nonnegatively weighted Robin boundary conditions and globalize by exhaustion of the regular locus.
Paper Structure (60 sections, 33 theorems, 153 equations)

This paper contains 60 sections, 33 theorems, 153 equations.

Table of Contents

  1. Introduction and statement of results
  2. Motivation: heat flow under SYZ-type collapse
  3. Geometric setting and analytic identifications
  4. Semi-flat control and vertical gap (interior).
  5. Normalized lift/average.
  6. Bilinear pairings and compressed semigroups
  7. Discriminant regime and intrinsic conic renormalization
  8. Conic model on the base.
  9. Why renormalization is necessary.
  10. Renormalized bilinear functional.
  11. Main result: renormalized bilinear limit across the discriminant
  12. Idea of the proof
  13. Further consequences and organization of the paper
  14. Semi-flat collapse and normalized identifications
  15. Geometric setting on the regular locus
  16. ...and 45 more sections

Key Result

Theorem 1.1

Assume the semi-flat control and the vertical spectral gap on every precompact subset of $B_{\mathrm{reg}}$, and assume that near each $p_j\in D$ the base metric $g_B$ is $C^1$-close to the flat cone $g_j^{\mathrm{cone}}$ with cone factor $\alpha_j$. Let $\Phi,\Psi\in C_c^\infty(B)$ and $\tau>0$. Th where $K_B^{\mathrm{ren}}$ is the canonical renormalized functional defined in eq:Kren-intro. If $\

Theorems & Definitions (74)

  • Theorem 1.1: Renormalized bilinear heat-kernel limit across the discriminant
  • Remark 1.2: Compressed semigroups and bilinear kernels
  • Remark 2.4: On rates
  • Lemma 2.5: Quantitative control of $(A_t,\rho_t)$
  • proof : Proof (sketch, with explicit dependencies)
  • Definition 2.6: Normalized lift and average
  • Lemma 2.7: Algebraic identities and adjointness
  • proof
  • Lemma 2.8: $L^2$ isometry and contraction
  • proof
  • ...and 64 more