Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Differential geometry via harmonic functions   In this talk, I will discuss the use of harmonic functions to study the
geometry and topology of complete manifolds. In my previous joint work with
Luen-fai Tam, we discovered that the number of infinities of a complete
manifold can be estimated by the dimension of a certain space of harmonic
functions. Applying this to a complete manifold whose Ricci curvature is almost
non-negative, we showed that the manifold must have finitely many ends. In my
recent joint works with Jiaping Wang, we successfully applied this general
method to two other classes of complete manifolds. The first class are
manifolds with the lower bound of the spectrum $\lambda_1(M) >0$ and whose
Ricci curvature is bounded by $$ Ric_M \ge -{m-2 \over m-1} \lambda_1(M). $$
The second class are stable minimal hypersurfaces in a complete manifold with
non-negative sectional curvature. In both cases we proved some splitting type
theorems and also some finiteness theorems.

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The categories of this paper are: mtrl-th
28
The categories of this paper are: alg-geom
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The categories of this paper are: q-alg
0
The categories of this paper are: hep-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: q-fin
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The categories of this paper are: bayes-an
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The categories of this paper are: funct-an
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The categories of this paper are: nucl-ex
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The categories of this paper are: astro-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: math-ph
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The categories of this paper are: patt-sol
1
The categories of this paper are: cs
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The categories of this paper are: econ
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The categories of this paper are: chao-dyn