Traces of Functions With Spacelike Graphs, and the Extension Problem Under Restrictions on the Gradient

Abstract

A 6-dimensional grand unified theory with the compact space having the topology of a real projective plane, i.e., a 2-sphere with opposite points identified, is considered. The space is locally flat except for two conical singularities where the curvature is concentrated. One supersymmetry is preserved in the effective 4d theory. The unified gauge symmetry, for example SU(5) , is broken only by the non-trivial global topology. In contrast to the Hosotani mechanism, no adjoint Wilson-line modulus associated with this breaking appears. Since, locally, SU(5) remains a good symmetry everywhere, no UV-sensitive threshold corrections arise and SU(5)-violating local operators are forbidden. Doublettriplet splitting can be addressed in the context of a 6d N = 2 super Yang-Mills theory with gauge group SU(6). If this symmetry is first broken to SU(5) at a fixed point and then further reduced to the standard model group in the above non-local way, the two light Higgs doublets of the MSSM are predicted by the group-theoretical and geometrical structure of the model.

On the uniqueness of the helicoid and Enneper’s surface in the Lorentz-Minkowski space $\mathbb \{R\}\textunderscore \{1\}^\{3\}$

“…where Ω is the domain bounded by π(γ) and d Ω is the inner distance in Ω (see [13]). Furthermore, S is a maximal surface provided that…”

“…Taking into account that M ∞ is the limit set of a sequence of embedded surfaces and that θ(s) is increasing, the sheets of the multigraph X : W → M ∞ ⊂ R 3 1 are ordered by height, i.e., (13) u…”

Extension of functions preserving the modulus of continuity

“…Unlike E1 and E2, the operator E is defined by f alone, without involving w. The operator E is well defined even if f is bounded but discontinuous on D. The following theorem provides an insight into the behavior of the function g = E(f) outside D. As an application of the extension operator E, let us consider the problem of the trace of a locally Lipschitz function first posed in [2].…”

A quasi-periodic minimal surface

“…For this we use the following theorem, which is a consequence of Theorem 1 in [5] and Theorem 4.1 in [2] :…”