Optimal bounds for ancient caloric functions

Dimensional bounds for ancient caloric functions on graphs

We study ancient solutions of polynomial growth to heat equations on graphs, and extend Colding and Minicozzi's theorem [CM19] 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.

体积倍增之外某些流形上的古代热函数空间

Space of ancient caloric functions on some manifolds beyond volume doubling

本文在突破体积倍增条件的情形下研究古代热函数空间，证明其时间多项式结构，并据此说明多项式增长调和函数有限性结果除多端情形外本质最优。

Ancient caloric functions on graphs with unbounded Laplacians

We study ancient solutions of polynomial growth to both continuous-time and discrete-time heat equations on graphs with unbounded Laplacians. We generalize Colding and Minicozzi's theorem [CM19] on manifolds, and the result [Hua19] on graphs with normalized Laplacians to the setting of graphs with unbounded Laplacians: For a graph admitting an intrinsic metric, which has polynomial volume growth, the dimension of the space of ancient solutions of polynomial growth is bounded by the dimension of harmonic functions with the same growth up to some factor.

四次热方程的古老解

Ancient Solutions to the Biharmonic Heat Equation

本文研究了四次热方程的古老解，并将其与多项式有界函数的空间维度联系起来，扩展了Colding和Minicozzi的研究。

梯度收缩里奇孤立子上多项式增长的调和函数与薛定谔函数

Harmonic and Schrödinger functions of polynomial growth on gradient shrinking Ricci solitons

本文在梯度收缩里奇孤立子上研究多项式增长的调和函数、古代热函数和薛定谔函数，在标量曲率至少二次衰减时证明调和函数与古代热函数空间的有限维性，并在无曲率条件下给出薛定谔函数空间的尖锐有限维估计。

Optimal curvature estimates for homogeneous Ricci flows

We prove uniform curvature estimates for homogeneous Ricci flows: For a solution defined on $[0,t]$ the norm of the curvature tensor at time $t$ is bounded by the maximum of $C(n)/t$ and $C(n) ( scal(g(t)) - scal(g(0)) )$. This is used to show that solutions with finite extinction time are Type I, immortal solutions are Type III and ancient solutions are Type I, where all the constants involved depend only on the dimension $n$. A further consequence is that a non-collapsed homogeneous ancient solution on a compact homogeneous space emerges from a unique Einstein metric on the same space. The a…