We obtain the evolution equations for the Riemann tensor, the Ricci tensor and the scalar curvature induced by the mean curvature flow. The evolution for the scalar curvature is similar to the Ricci flow, however, negative, rather than positive, curvature is preserved. Our results are valid in any dimension.

We study strongly coupled mass-deformed-CFT on a fixed de Sitter spacetime in three dimensions via holography. elucidate the global causal structure of four-dimensional dual to invariant vacuum state. The conformal boundaries appear as spacelike defects sourced by mass deformation, which extend into bulk curvature singularities AdS. compute all one- and two-point functions deformed-CFT stress t...

This article investigates stationary surfaces with boundaries, which arise as the critical points of functionals dependent on curvature. Precisely, a generalized “bending energy” functional $$\mathcal {W}$$ is considered involves Lagrangian that symmetric in principal curvatures. The first variation computed, and stress tensor extracted whose divergence quantifies deviation from -criticality. B...

The regularized expectation value of the stress-energy tensor for a massless bosonic or fermionic field in 1+1 dimensions is calculated explicitly for the instantaneous vacuum relative to any Cauchy surface (here a spacelike curve) in terms of the length L of the curve (if closed), the local extrinsic curvature K of the curve, its derivative K ′ with respect to proper distance x along the curve...

Tensor voting is an efficient algorithm for perceptual grouping and feature extraction, particularly for contour extraction. In this paper two studies on tensor voting are presented. First the use of iterations is investigated, and second, a new method for integrating curvature information is evaluated. In opposition to other grouping methods, tensor voting claims the advantage to be non-iterat...

Motivated by the strong maximum principle for Paneitz operator in dimension 5 or higher found in [GM] and the calculation of the second variation of the Green’s function pole’s value on S3 in [HY2], we study Riemannian metric on 3 manifolds with positive scalar and Q curvature. Among other things, we show it is always possible to …nd a constant Q curvature metric in the conformal class. Moreove...

We investigate the geometry of a twisting non-shearing congruence null geodesics on conformal manifold even dimension greater than four and Lorentzian signature. give necessary sufficient condition Weyl tensor for twist to induce an almost Robinson structure, that is, screen bundle is equipped with complex structure. In this case, (local) leaf space acquires partially integrable contact CR stru...