One of the earliest applications of modern integrable systems theory (or “soliton theory”) to differential geometry was the solution of the problem of finding all constant mean curvature (CMC) tori in R3 (and therefore, by taking the Gauss map, finding all non-conformal harmonic maps from a torus to S2). At its simplest level this proceeds from the recognition that the Gauss-Codazzi equations o...

We study static black hole solutions in Einstein and Einstein–Gauss–Bonnet gravity with the topology of the product of two spheres,Sn × Sn, in higher dimensions. There is an unusual new feature of the Gauss–Bonnet black hole: the avoidance of a non-central naked singularity prescribes a mass range for the black hole in terms of > 0. For an Einstein–Gauss–Bonnet black hole a limited window of ne...

We study the evolution of convex complete non-compact graphs by positive powers Gauss curvature. show that if initial graph has a local uniform convexity, then evolves any power curvature for all time. In particular, is not necessarily differentiable.

The isometric immersion of two-dimensional Riemannian manifolds or surfaces with negative Gauss curvature into the three-dimensional Euclidean space is studied in this paper. The global weak solutions to the Gauss-Codazzi equations with large data in L∞ are obtained through the vanishing viscosity method and the compensated compactness framework. The L∞ uniform estimate and H−1 compactness are ...

In 1963, K.P. Grotemeyer proved an interesting variant of the Gauss-Bonnet Theorem. Let M be an oriented closed surface in the Euclidean space R 3 with Euler characteristic χ(M), Gauss curvature G and unit normal vector field n. Grote-meyer's identity replaces the Gauss-Bonnet integrand G by the normal moment (a · n) 2 G, where a is a fixed unit vector: M (a · n) 2 Gdv = 2π 3 χ(M). We generaliz...

In this paper I will provide a proof of the Gauss-Bonnet Theorem. I will start by briefly explaining regular surfaces and move on to the first and second fundamental forms. I will then discuss Gaussian curvature and geodesics. Finally, I will move on to the theorem itself, giving both a local and a global version of the Gauss-Bonnet theorem. For this paper, I will assume that the reader has a k...

The paper concerns the problem of correct curvatures estimates directly from polygonal meshes. We present a new algorithm that allows the construction of unambiguous Gauss maps for a large class of polyhedral surfaces, including surfaces of non-convex objects and even non-manifold surfaces. The resulting Gauss map provides shape recognition and curvature characterisation of the polyhedral surfa...