## Heat kernel upper bounds on a complete non-compact manifold.(English)Zbl 0810.58040

Let $$\Lambda (\nu)$$ be a positive continuous decreasing function defined for $$\nu \in {\mathbf R}_ +$$ and let $$M$$ be a smooth connected noncompact geodesically complete Riemannian manifold of dimension $$n \geq 2$$. One says that a $$\Lambda$$-isoperimetric inequality is valid for a domain $$\Omega \subset M$$ if, for any subdomain $$D \subset \Omega$$ the first Dirichlet eigenvalue $$\lambda_ 1 (M)$$ for the Laplacian on $$M$$ is bounded below by $$\Lambda (\text{Vol }D)$$. The author’s main result is that the existence of such a $$\Lambda$$-isoperimetric inequality for a precompact domain $$\Omega \subset M$$ implies a strong upper bound for the heat kernel $$p(x,y, t)$$ on $$M$$, which in turn implies a lower bound for the $$k$$-th eigenvalue of the Laplacian on $$M$$. The upper bound for the heat kernel is in terms of a function $$V(t)$$ defined by $$t = \int_ 0^{V(t)^ -} (\nu \Lambda(\nu))^{-1} d\nu$$. Much of the work involves the growth properties of the function $$V$$. The author’s estimate $$p(x, y, t) \leq C(V(ct))^{-1} \exp (-r^ 2/Dt)$$ for $$t > 0$$, positive constants $$c$$, $$C$$ and $$D > 4$$ arbitrarily close to 4, includes many previously known estimates, including those of Varopoulos, Cheng, Li and Yau, and Davies. For further details see the paper.
Reviewer: J.S.Joel (Kelly)

### MSC:

 58J35 Heat and other parabolic equation methods for PDEs on manifolds 35P15 Estimates of eigenvalues in context of PDEs 58J50 Spectral problems; spectral geometry; scattering theory on manifolds

### Keywords:

heat kernel; isoperimetric inequalities; spectral geometry
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