# The Dirichlet problem for constant mean curvature surfaces in Heisenberg space

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The hypothesis of local compactness of the target is removed from an earlier result about interior HÃ¶lder continuity of locally energy minimizing maps Ï• from a Riemannian polyhedron ( X, g) to a suitable ball B of radius R < Ï€/2 (best possible) in a geodesic space with curvature...

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Consider a compact Riemannian manifold M of dimension n whose boundary ?M is totally geodesic and is isometric to the standard sphere S. A natural conjecture of Min-Oo asserts that if the scalar curvature of M is at least n( n-1), then M is isometric to the hemisphere $S_{+}^{n}$ equipped with...

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We study shapes of planar arcs and closed contours modeled on elastic curves obtained by bending, stretching or compressing line segments non-uniformly along their extensions. Shapes are represented as elements of a quotient space of curves obtained by identifying those that differ by...

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We show that all geodesic rays in the uniform infinite half-planar quadrangulation (UIHPQ) intersect the boundary infinitely many times, answering thereby a recent question of Curien. However, the possible intersection points are sparsely distributed along the boundary. As an intermediate step,...

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We describe some results on the exact boundary controllability of the wave equation on an orientable two-dimensional Riemannian manifold with nonempty boundary. If the boundary has positive geodesic curvature, we show that the problem is controllable in finite time if (and only if) there are no...