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Unit# The Troyanov Group

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Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Rieman

Riemannian manifold

In differential geometry, a Riemannian manifold or Riemannian space (M, g), so called after the German mathematician Bernhard Riemann, is a real, smooth manifold M equipp

Metric space

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function.

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For sequences of warped product metrics on a 3-torus satisfying the scalar curvature bound Rj = -1j, uniform upper volume and diameter bounds, and a uniform lower area bound on the smallest minimal surface, we find a subsequence which converges in both the Gromov-Hausdorff and the Sormani-Wenger intrinsic flat sense to a flat 3-torus.

Luigi Provenzano, Joachim Stubbe

We present upper and lower bounds for Steklov eigenvalues for domains in RN+1 with C-2 boundary compatible with the Weyl asymptotics. In particular, we obtain sharp upper bounds on Riesz-means and the trace of corresponding Steklov heat kernel. The key result is a comparison of Steklov eigenvalues and Laplacian eigenvalues on the boundary of the domain by applying Pohozaev-type identities on an appropriate tubular neigborhood of the boundary and the min-max principle. Asymptotically sharp bounds then follow from bounds for Riesz-means of Laplacian eigenvalues.

Luigi Provenzano, Joachim Stubbe

We present asymptotically sharp inequalities, containing a 2nd term, for the Dirichlet and Neumann eigenvalues of the Laplacian on a domain, which are complementary to the familiar Berezin-Li-Yau and Kroger inequalities in the limit as the eigenvalues tend to infinity. We accomplish this in the framework of the Riesz mean R-1(z) of the eigenvalues by applying the averaged variational principle with families of test functions that have been corrected for boundary behaviour.