熱方程的遠古解問題為數學中一個重要的主題。首先，我們聚焦在完備黎曼流行上的拉普拉斯-貝爾特拉米算子並學習多項式增長之調和函數空間的維度估計。其次，我們研究多項式增長之熱方程原古解的空間並估計其維度。最後，我們研究完備黎曼流行上霍奇-拉普拉斯算子對應的熱方程之解。在具有多項式體積增長的完備黎曼流行上，我們給出多項式增長熱流調和形式的原古解之空間的維度估計，其維度估計的上界和相對應的調和形式空間之維度相關。

The study of ancient solutions to the heat equation, as a prototypical parabolic partial differential equation, is one of important subjects in mathematics.
We first focus on the Laplace-Beltrami operator on complete Riemannian manifolds and study the dimension estimate for the space of harmonic functions with polynomial growth. Secondly, we study the space of ancient solution to the heat equation with polynomial growth and give the dimension estimate on such a space. Finally, we study the heat equation associated to the Hodge Laplace operator on complete Riemannian manifolds. If the manifold has polynomial volume growth, we have the dimension estimate for the space of the polynomial growth ancient solutions to harmonic form heat flow. Such a dimension is bounded by that of the space of harmonic forms.

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