The Higgs mechanism and geometrical flows for two-manifolds

Using Perelman’s approach for geometrical flows in terms of an entropy functional, the Higgs mechanism is studied dynamically along flows defined in the space of parameters and in fields space. The model corresponds to two-dimensional gravity that incorporates torsion as the gradient of a Higgs field, and with the reflection symmetry to be spontaneously broken. The results show a discrete mass spectrum and the existence of a mass gap between the Unbroken Exact Symmetry and the Spontaneously Broken Symmetry scenarios. In the latter scenario, the geometries at the degenerate vacua correspond to conformally flat manifolds without torsion; twisted two-dimensional geometries are obtained by building perturbation theory around a ground state; the tunneling quantum probability between vacua is determined along the flows.

On n-superharmonic functions and some geometric applications

In this paper we study asymptotic behaviors of n-superharmonic functions at singularity using the Wolff potential and capacity estimates in nonlinear potential theory. Our results are inspired by and extend [6] of Arsove–Huber and [63] of Taliaferro in 2 dimensions. To study n-superharmonic functions we use a new notion of thinness in terms of n-capacity motivated by a type of Wiener criterion in [6]. To extend [63], we employ the Adams–Moser–Trudinger’s type inequality for the Wolff potential, which is inspired by the inequality used in [15] of Brezis–Merle. For geometric applications, we study the asymptotic end behaviors of complete conformally flat manifolds as well as complete properly embedded hypersurfaces in hyperbolic space. These geometric applications seem to elevate the importance of n-Laplace equations and make a closer tie to the classic analysis developed in conformal geometry in general dimensions.

Prescribed Schouten Tensor in Locally Conformally Flat Manifolds

We consider the pseudo-Euclidean space $$({\mathbb {R}}^n,g)$$(Rn,g), with $$n \ge 3$$n≥3 and $$g_{ij} = \delta _{ij} \varepsilon _{i}$$gij=δijεi, where $$\varepsilon _{i} = \pm 1$$εi=±1, with at least one positive $$\varepsilon _{i}$$εi and non-diagonal symmetric tensors $$T = \sum \nolimits _{i,j}f_{ij}(x) dx_i \otimes dx_{j} $$T=∑i,jfij(x)dxi⊗dxj. Assuming that the solutions are invariant by the action of a translation $$(n-1)$$(n-1)- dimensional group, we find the necessary and sufficient conditions for the existence of a metric $$\bar{g}$$g¯ conformal to g, such that the Schouten tensor $$\bar{g}$$g¯, is equal to T. From the obtained results, we show that for certain functions h, defined in $$\mathbb {R}^{n}$$Rn, there exist complete metrics $$\bar{g}$$g¯, conformal to the Euclidean metric g, whose curvature $$\sigma _{2}(\bar{g}) = h$$σ2(g¯)=h.