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It is a classical Sobolev inequality that if $\phi$ is an eigenfunction of the Laplace-Beltrami operator on a $n$-dim compact Riemannian manifold $M$ with eigenvalue $\lambda$ then $$||\phi||_{L^\infty(M)}\le c(n)\lambda^{\frac{n-1}{4}}||\phi||_{L^2(M)}.$$ It is a theorem that (possibly by P.H.Berard) that of sectional curvature of $M$ is strictly negative then the above bound can be improved in terms of $\lambda$ using dynamic properties of $M$ (ergodicity of geodesic flow etc.)

I am not sure what exactly the result is (in a talk I heard that there will be a saving of logarithm of $\lambda$). Could someone please exactly describe the result? Also please provide a reference to that as I am unable to find that.

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  • $\begingroup$ Yes, there is a log improvement for $L^{\infty}$ norm of the eigenfunction for manifolds with negative curvature. Actually, it's proved recently that there is also a log improvement for $L^p$ norm with $p>\frac{2(n+1)}{n-1}$. The classical $L^p$ norm s(including $p=\infty$)for general manifolds was first proved by C.D.Sogge, and the result is sharp by testing spherical harmonics. You can find this in his book "Fourier integrals in classical analysis". $\endgroup$
    – Tomas
    Commented Apr 15, 2015 at 11:53
  • $\begingroup$ Could you please tell me the exact statement of Berard's theorem of $\log$ saving? $\endgroup$
    – Subhajit Jana
    Commented Apr 15, 2015 at 19:45
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    $\begingroup$ A nice reference is Sogge's latest book Hangzhou Lectures on Eigenfunctions of the Laplacian (Princeton Univ. Press, 2014). $\endgroup$
    – ifw
    Commented Apr 16, 2015 at 11:41

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