positive non-constant harmonic function $f $ in $L^1(M)$ on a complete Riemannian manifold

Question

Let $M$ be a complete Riemannian manifold, does there exists a positive non-constant harmonic function $f \in L^1(M)$? Who can answer me or give me a counter example? Thank you very much!

Answer 1

Consider the surface of revolution (so topologically we are dealing with $\mathbb{R}\times\mathbb{S}$) with standard coordinates $(z,\theta)$. Let the metric be $$ \mathrm{d}s^2 = \mathrm{d}z^2 + h^2(z) \mathrm{d}\theta^2$$ This manifold is clearly complete (it is a warped product of two geodesically complete manifolds).

The Laplace-Beltrami operator associated to it is $$ \triangle = \frac{1}{h} \partial_z h \partial_z + \frac{1}{h^2} \partial^2_\theta $$ and the volume/area form is $h \mathrm{d}z \mathrm{d}\theta$.

Let $f = f(z)$ be a function. It being $L^1(M)$ is equivalent to $$ \int_{-\infty}^\infty |f(z)| h(z) \mathrm{d}z < \infty $$ It being harmonic is the same as $$ h \partial_z f \equiv c $$ for some constant $c$ (which we can assume, WLOG, to be 1). So this implies that we need to find a monotonic function $f$ such that $f / f'$ is absolutely integrable. This requires that $\frac{d}{dz} \log f$ to grow superlinearly in $z$.

So we can consider the following: let $f(z) = \exp (z + z^3)$. Define $h(z) = \frac{1}{(1 + 3z^2) \exp (z + z^3)}$. Then $h f' = 1$ so $f$ is harmonic. On the other hand, $hf = \frac{1}{1+3z^2}$ is integrable in $z$, and hence $f\in L^1(M)$.

Note that the scalar curvature can be computed to be $$ - \frac{2}{h} h''$$ which has fourth order growth in $z$ and so violates the hypotheses of Li's theorem. (In fact, the order of growth of the scalar curvature will be roughly twice that of the growth of $\frac{d}{dz} \log f$. So in this sense the quadratic growth assumption in Li's theorem is sharp.)

Answer 2

No.

The second result on Google gives http://intlpress.com/JDG/archive/1984/20-2-447.pdf, "Uniqueness of $L^1$ solutions for the Laplace equation and heat equation on Riemannian manifolds" by Peter Li, J Diff Geo 20 (1984) 447-457. It has the following result:

Theorem 1: If $M$ is a complete noncompact Riemannian manifold without boundary, and if the Ricci curvature of $M$ has a negative quadratic lower bound, then any $L^1$ nonnegative subharmonic function on $M$ is identically constant. In particular, any $L^1$ nonnegative harmonic function on $M$ is identically constant.