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In this paper we prove that given a volume, among all domains with smoothboundary in rank-1 symmetric spaces of noncompact type, geodesic ballsmaximizes the first nonzero Steklov eigenvalue. We also prove a comparisonresult for the first nonzero Steklov eigenvalue for domains in simply connectedRiemannian manifolds with certain curvature bounds.

Sharp upper bound and a comparison theorem for the first nonzero Steklov eigenvalue