# Dimensional Bounds for Ancient Caloric Functions on Graphs

@article{Hua2019DimensionalBF,
title={Dimensional Bounds for Ancient Caloric Functions on Graphs},
author={Bobo Hua},
journal={International Mathematics Research Notices},
year={2019}
}
• B. Hua
• Published 6 March 2019
• Mathematics
• International Mathematics Research Notices
We study ancient solutions of polynomial growth to heat equations on graphs and extend Colding and Minicozzi’s theorem [9] on manifolds to graphs: for a graph of polynomial volume growth, the dimension of the space of ancient solutions of polynomial growth is bounded by the product of the growth degree and the dimension of harmonic functions with the same growth.
8 Citations
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• 2019
It is well known that generic solutions of the heat equation are not analytic in time in general. Here it is proven that ancient solutions with exponential growth are analytic in time in ${\M} \times Time analyticity for the heat equation and Navier-Stokes equations • Mathematics Journal of Functional Analysis • 2020 Time Analyticity for Inhomogeneous Parabolic Equations and the Navier–Stokes Equations in the Half Space • Mathematics • 2020 We prove the time analyticity for weak solutions of inhomogeneous parabolic equations with measurable coefficients in the half space with either the Dirichlet boundary condition or the conormal ## References SHOWING 1-10 OF 30 REFERENCES Optimal bounds for ancient caloric functions • Mathematics Duke Mathematical Journal • 2021 For any manifold with polynomial volume growth, we show: The dimension of the space of ancient caloric functions with polynomial growth is bounded by the degree of growth times the dimension of Polynomial growth harmonic functions on groups of polynomial volume growth • Mathematics • 2012 We consider harmonic functions of polynomial growth of some order $$d$$d on Cayley graphs of discrete groups of polynomial volume growth of order $$D$$D w.r.t. the word metric and prove the optimal Geometric analysis aspects of infinite semiplanar graphs with nonnegative curvature II • Mathematics • 2011 In the present paper, we apply Alexandrov geometry methods to study geometric analysis aspects of infinite semiplanar graphs with nonnegative combinatorial curvature in the sense of Higuchi. We Harmonic sections of polynomial growth defined on M which has at most polynomial growth of degree d must be finite dimensional for any d ∈ R. In fact, Colding and Minicozzi [C-M 4] announced, and subsequently proved in [C-M 6], a more Volume doubling, Poincaré inequality and Guassian heat kernel estimate for nonnegative curvature graphs • Mathematics • 2014 By studying the heat semigroup, we prove Li-Yau type estimates for bounded and positive solutions of the heat equation on graphs, under the assumption of the curvature-dimension inequality Bounding dimension of ambient space by density for mean curvature flow For an ancient solution of the mean curvature flow, we show that each time slice Mt is contained in an affine subspace with dimension bounded in terms of the density and the dimension of the evolving On Ancient Solutions of the Heat Equation • Mathematics Communications on Pure and Applied Mathematics • 2019 An explicit representation formula with Martin boundary for all positive ancient solutions of the heat equation in the euclidean case is found. In the Riemannian case with nonnegative Ricci Disorder, entropy and harmonic functions • Mathematics • 2011 We study harmonic functions on random environments with particular emphasis on the case of the infinite cluster of supercritical percolation on$\mathbb{Z}^d\$. We prove that the vector space of
A new proof of Gromov's theorem on groups of polynomial growth
We give a new proof of Gromov’s theorem that any finitely generated group of polynomial growth has a finite index nilpotent subgroup. The proof does not rely on the MontgomeryZippin-Yamabe structure
Polynomials and harmonic functions on discrete groups
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• 2015
Alexopoulos proved that on a finitely generated virtually nilpotent group, the restriction of a harmonic function of polynomial growth to a torsion-free nilpotent subgroup of finite index is always a