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Skip to search form Skip to main content You are currently offline. Some features of the site may not work correctly. DOI: Li and S. Yau Published Mathematics Acta Mathematica. Estimations de gradients. Inegalites de Harnack. Majorations et minorations des solutions fondamentales. Equation de la chaleur et noyau de Green. Operateur de Schrodinger. View on Springer.
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Highly Influenced. View 7 excerpts, cites background. Comparison theorems in Riemannian geometry. Highly Influential. View 2 excerpts, references background. Instantons, double wells and large deviations. View 1 excerpt, references background. The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature. Curvature, diameter and betti numbers. The Poisson Kernel on positively curved manifolds.Exploiting this concept, we study submersions arising from actions of Lie groups.
In this context, we extend the state-of-the-art results on the bottom of the spectrum under Riemannian coverings. As an application, we compute the bottom of the spectrum and the Cheeger constant of connected, amenable Lie groups. The study of the spectrum of the Laplacian on a Riemannian manifold has attracted much attention over the last years.
In order to comprehend its relations with the geometry of the underlying manifold, it is reasonable to investigate its behavior under maps between Riemannian manifolds that respect the geometry of the manifolds to some extent. In this paper, we study the behavior of the spectrum under Riemannian submersions. The notion of Riemannian submersion was introduced in the s as a tool to express the geometry of a manifold in terms of the geometry of simpler components, namely, the base space and the fibers.
Of course, by geometry of the fibers, we mean both their intrinsic and their extrinsic geometry as submanifolds of the total space. Bearing this in mind, it is natural to describe the spectrum of the total space in terms of the geometry and the spectrum of the base space and the fibers.
More precisely, according to [ 23Theorem 1. In the second part of [ 23 ], following [ 4 ], we studied Riemannian submersions with closed fibers. It should be noticed that if the submersion has fibers of infinite volume, then we are not able to define that operator, at least in the way we did in [ 23 ].
Furthermore, it is evident that S coincides with the Laplacian, if the submersion has minimal fibers. It should be emphasized that no assumptions on the geometry or the topology of the manifolds are required in this theorem. In particular, the manifolds do not have to be complete. This, together with the decomposition principle, allows us to derive a similar inequality involving the bottoms of the essential spectra, if the fibers are closed.
Thus, Theorem 1. For Riemannian submersions with closed fibers, we obtain the following consequence of Theorem 1.
This equivalence has been extended in [ 23Corollary 1. Corollary 1.
If, in addition, the manifolds involved in Corollary 1. This, together with Theorem 1. In the second part of the paper, we study Riemannian principal bundles. We then say that p is a Riemannian submersion arising from the action of G. The behavior of the spectrum under such submersions has been studied for instance in [ 13 ].
In the case where G is a discrete group, its action gives rise to a normal Riemannian covering. In this context, there are various results establishing relations between properties of the deck transformation group and the behavior of the spectrum. Brooks was the first one to investigate when the equality holds. Theorem 1. Recall that there exist connected Lie groups that are amenable but not unimodular because any solvable group is amenableand connected Lie groups that are unimodular but not amenable since any connected, semisimple Lie group is unimodular.
It is notable that if G is compact, then Corollary 1. Even though Theorem 1. We will construct a wide class of examples demonstrating that this assumption is essential. This is because in Theorem 1.
Finally, exploiting Theorems 1. In this setting, we obtain some relations between the mean curvature of the subgroup and the bottom of the spectrum of the group, the subgroup, and the quotient.
Let G be a connected Lie group endowed with a left-invariant metric and N be a closed as a subsetconnected, normal subgroup of G with mean curvature H. As an application of this theorem, we compute the bottom of the spectrum and the Cheeger constant of connected, amenable Lie groups.Thanks for helping us catch any problems with articles on DeepDyve.
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Kernel estimates for Schrödinger type operators with unbounded coefficients and critical exponents
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Pallara and A. Rhandi J. Section 4. The behaviour of c t near 0 is also shown to be precise. Similar bounds are also proved for the derivatives of p. Our analysis provides a family of such estimates e. Estimates of our form for the heat kernel of more general even nonsymmetric operators have been obtained recently in  using ultracontractivity methods in weighted spaces.
In particular, we show that our Example 3. Due to the behaviour of c t near 0, however, the upper bound of Example 3. We also refer the reader to the papers  and  for bounds on the heat kernels associated with potentials of polynomial type, different from ours. In particular, potentials not tending to infinity are treated in  and lower bounds can be found in . The plan of the paper is the following. We show how the integrability of certain unbounded functions with respect to p x, y, t dy can be obtained via Lyapunov function techniques.
This allows us to indicate some growth conditions on the potential V which imply precise estimates of the L1 -norm of Lyapunov functions with respect to the kernel p.We prove sharp point-wise estimates for the associated semigroups which show, in particular, how the boundary conditions affect the time decay of the heat kernel in dimensions one and two.
Applications to spectral estimates are discussed as well. This is a preview of subscription content, access via your institution.
Rent this article via DeepDyve. Adams, R. Elsevier, UK Google Scholar.Evgeny Korotyaev, Inverse problems and estimates for Schrödinger operators on the circle
Arendt, W. In: Dafermos, C. Handbook of Differential Equations: Evolutionary Equations, vol. Brezis, H. Springer-Verlag, Berlin Davies, E. B: Spectral Theory and Differential Operators. Cambridge University Press B: Heat Kernels and Spectral Theory. Ekholm, T.
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