# Convergence of heat flow on non-compact manifolds?

Consider the heat equation $$\partial_t u= \Delta u+\lambda_1 u$$ on a non-compact complete manifold $$M$$ (with nonpositive curvature) where $$\lambda_1$$ is the first eigenvalue and we start with some smooth initial data $$u_0$$ at $$t=0.$$ Then does the heat flow converge to the first eigenfunction on $$M$$?

• eigenfunction/eigenvalue in what sense? $L^2$? Apr 17 at 20:46
• yeah, that's right Apr 17 at 20:46
• Convergence in which sense? $L^2$-convergence follows from the spectral theorem. Do you want something stronger? Apr 19 at 9:34
• @MaoWao I'd love to see a proof of that in an answer, is the convergence rate exponential? Does one also have convergence $C^0$ on compact subsets? Apr 19 at 20:33
• @Overflowian See my answer. Regarding exponential decay, an estimate of the form $\lVert u_t-u_\infty\rVert_2\leq e^{-\alpha t}\lVert u_0-u_\infty\rVert_2$ is equivalent to a gap of size at least $\alpha$ between $\lambda_1$ and the rest of the spectrum. Apr 20 at 6:44

The following answer only applies if $$u_0\in L^2$$, and it mostly relies on abstract operator theoretic arguments. I am sure more could be said by exploiting the geometric structure.

The Laplace-Beltrami operator on a complete manifold is essentially self-adjoint on $$C_c^\infty$$. Since $$\lambda_1$$ is the bottom of the spectrum of $$-\Delta$$, the operator $$-(\Delta+\lambda_1)$$ has a unique positive self-adjoint extension, which I denote by $$A$$. The heat flow is given by $$e^{-tA}u_0$$.

Let $$E$$ denote the spectral measure of $$A$$ and $$d\mu(\lambda)=d\langle u_0,E(\lambda) u_0\rangle$$. Note that $$\mu$$ is a finite measure on $$\mathbb R_+$$ with $$\mu(\mathbb R_+)=\lVert u_0\rVert_2^2$$. By the spectral theorem we have $$\lVert A^k(e^{-tA}-1_{\{0\}}(A))u_0\rVert_2^2=\int_{[0,\infty)}\lambda^{2k}(e^{-t\lambda}-1_{\{0\}}(\lambda))^2\,d\mu(\lambda).$$ For $$t\geq t_0$$ (and $$\lambda\geq 0$$) we have $$\lambda^{2k}(e^{-t\lambda}-1_{\{0\}}(\lambda))^2\leq \lambda^{2k}e^{-2t\lambda}\lesssim_k t_0^{-2k}.$$ Thus $$\lVert A^k(e^{-tA}-1_{\{0\}}(A))u_0\rVert_2^2\to 0$$ as $$t\to\infty$$ by the dominated convergence theorem.

The operator $$1_{\{0\}}(A)$$ is the projection onto $$\ker(\Delta+\lambda_1)$$. In particular, if $$\lambda_1$$ is an eigenvalue of $$-\Delta$$, then $$1_{\{0\}}(A)u_0$$ is an eigenfunction of $$-\Delta$$. In the following I will write $$u_t$$ for $$e^{-tA}u_0$$ and $$u_\infty$$ for $$1_{\{0\}}(A)u_0$$. By what we have seen before, $$u_t\to u_\infty$$ in $$L^2$$.

By elliptic regularity and Sobolev embedding, if $$m\in\mathbb N$$ and $$2k\geq m+\dim(M)/2+1$$, then $$\lVert u_t-u_\infty\rVert_{C^m(\Omega)}\lesssim_{m,\Omega}\lVert u_t-u_\infty\rVert_{H^{2k}(\Omega)}\lesssim_{k,\Omega}\lVert A^k(u_t-u_\infty)\rVert_{L^2(\Omega)}+\lVert u_t-u_\infty\rVert_{L^2(\Omega)}$$ for every sufficiently nice bounded domain $$\Omega$$ in $$M$$.

Therefore, $$u_t\to u_\infty$$ in $$C^m$$ on compact subsets for every $$m\in\mathbb N$$.

• I'm a bit confused by signs here, you seems to say that the spectrum of $-\Delta$ satisfies $\sigma(-\Delta)\subset (\lambda_1,+\infty)$. So is the definition of $\lambda_1$ given by $\inf \sigma(-\Delta)$?. I thought that it was the smallest positive eigenvalue and in particular it should be contained in the spectrum. Apr 20 at 12:05
• I took $\lambda_1$ to mean the bottom of the spectrum of $-\Delta$ (which is actually contained in the spectrum because it is closed). Of course, if $\lambda_1$ is embedded in the spectrum, this is no longer true -- then you can have approximate eigenfunctions of the bottom of the spectrum for which $e^{t(\Delta+\lambda_1)}u_0$ blows up in $L^2$ norm. Apr 20 at 12:24
• So $-\Delta-\lambda_1$ is positive semi-definite, does it still have a unique extension? Apr 20 at 12:28