Jiaping Wang

Spaces of Harmonic Functions

It is important and interesting to study harmonic functions on a Riemannian manifold. In an earlier work of Li and Tam [ 21 ] it was demonstrated that the dimensions of various spaces of bounded and positive harmonic functions are closely related to the number of ends of a manifold. For the linear space consisting of all harmonic functions of polynomial growth of degree at most d on a complete Riemannian manifold M n of dimension n , denoted by [Hscr ] d ( M n ), it was proved by Li and Tam [ 20 ] that the dimension of the space [Hscr ] 1 ( M ) always satisfies dim[Hscr ] 1 ( M ) [les ] dim[Hscr ] 1 (ℝ n ) when M has non-negative Ricci curvature. They went on to ask as a refinement of a conjecture of Yau [ 32 ] whether in general dim [Hscr ] d ( M n ) [les ] dim[Hscr ] d (ℝ n ) for all d . Colding and Minicozzi made an important contribution to this question in a sequence of papers [ 5–11 ] by showing among other things that dim[Hscr ] d ( M ) is finite when M has non-negative Ricci curvature. On the other hand, in a very remarkable paper [ 16 ], Li produced an elegant and powerful argument to prove the following. Recall that M satisfies a weak volume growth condition if, for some constant A and ν, formula here for all x ∈ M and r [les ] R , where V x ( r ) is the volume of the geodesic ball B x ( r ) in M ; M has mean value property if there exists a constant B such that, for any non- negative subharmonic function f on M , formula here for all p ∈ M and r > 0.

Geometry of shrinking Ricci solitons

The main purpose of this paper is to investigate the curvature behavior of four dimensional shrinking gradient Ricci solitons. For such soliton

Counting dimensions of L-harmonic functions

M

Bounded harmonic maps on a class of manifolds

with bounded scalar curvature

The heat flow and harmonic maps between complete manifolds

S

Conical structure for shrinking Ricci solitons

, it is shown that the curvature operator

Polynomial growth solutions of uniformly elliptic operators of non-divergence form

\mathrm{Rm}

Spectral gap estimates on compact manifolds

of

Ends of locally symmetric spaces with maximal bottom spectrum

M

Holomorphic functions on Kähler–Ricci solitons

satisfies the estimate