In this work 1 the curvature of Weinhold (thermodynamical) metric is studied in the case of systems with two thermodynamical degrees of freedom. Conditions for the Gauss curvature R to be zero, positive or negative are worked out. Signature change of the Weinhold metric and the corresponding singular behavior of the curvature at the phase boundaries are studied. Cases of systems with the consta...

We relate the total curvature and the isoperimetric deficit of a curve γ in a two-dimensional space of constant curvature with the area enclosed by the evolute of γ. We provide also a Gauss-Bonnet theorem for a special class of evolutes.

In this work, we give a priori height and gradient estimates for solutions of the prescribed constant Gauss curvature equation in Euclidean space. We shall consider convex radial graphs with positive constant mean curvature. The estimates are established by considering in such a graph, the Riemannian metric given by the second fundamental form of the immersion. r 2003 Elsevier Inc. All rights r...

Abstract. On the basis of the second gradient operator defined on curved surfaces, the second category of integral theorems for tensor fields, including the second divergence theorems, the second gradient theorems, the second curl theorems and the second circulation theorems, are systematically demonstrated. Simple conservation laws about the mean curvature and Gauss curvature are deduced from ...

We study the nodal sets of eigenfunctions of the Laplacian on the standard d-dimensional flat torus. The question we address is: Can a fixed hypersurface lie on the nodal sets of eigenfunctions with arbitrarily large eigenvalue? In dimension two, we show that this happens only for segments of closed geodesics. In higher dimensions, certain cylindrical sets do lie on nodal sets corresponding to ...

We prove that convex hypersurfaces in Rn+1 contracting under the flow by any power α > 1 n+2 of the Gauss curvature converge (after rescaling to fixed volume) to a limit which is a smooth, uniformly convex self-similar contracting solution of the flow. Under additional central symmetry of the initial body we prove that the limit is the round sphere.

1. Summary of results. The following is known: let 5 be a minimal surface defined by z=f(x, y) over the region D:x2+y2<R2, and let p be the point of S over the origin. Let W= (1+fl+fl)112 at p. Then the Gauss curvature K at p satisfies \K\ Sc/R2W2. The best numerical value of c known previously was 12.25. This inequality is simultaneously sharpened and generalized. First of all, it is proved th...

The mathematicians for centuries have researched the surfaces theory. In this paper, we consider strophoidal surface in three dimensional Euclidean space E^3. We present notations of a geometry. addition, stating helicoidal surface, define and calculate its Gauss map, Gaussian curvature, mean curvature. Finally, give some relations curvature that kind surfaces.

We study the cosmology of the Randall-Sundrum brane-world where the Einstein-Hilbert action is modified by curvature correction terms: a four-dimensional scalar curvature from induced gravity on the brane, and a five-dimensional Gauss-Bonnet curvature term. The combined effect of these curvature corrections to the action removes the infinite-density big bang singularity, although the curvature ...

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...