The modulus of the annulus between two nested curves is monotonically increasing under curve shortening flow in the plane and on surfaces with curvature bounds.
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The first correction term in the mean first capture time of shrinking geodesic balls by isotropic Lévy flights on Zoll surfaces is determined by the degree of the conjugate point.
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Monotonicity of the modulus under curve shortening flow
The modulus of the annulus between two nested curves is monotonically increasing under curve shortening flow in the plane and on surfaces with curvature bounds.
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Geodesic L\'evy flights on Zoll surfaces
The first correction term in the mean first capture time of shrinking geodesic balls by isotropic Lévy flights on Zoll surfaces is determined by the degree of the conjugate point.