Heat Flow under Semi-Flat Collapse with Conic Renormalization

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.