We discuss several structural properties of functions belonging to a parabolic energy class, reminiscent of the elliptic De Giorgi class. In earlier works, sub-potential lower bounds, giving insight into the structural behavior of elements of these classes, were established for the linear case: Here, we extend these results to the nonlinear one. By showing that sub-potential lower bounds follow solely from the Harnack inequality, we show that positive solutions to Trudinger’s equation and elements of parabolic De Giorgi classes have a common lower bound. For both cases, we derive Liouville-type rigidity results in the parabolic setting.
Normal functions v provide a method for studying algebraic cycles Z(t) subset of X-t varying in a family of smooth projective varieties. Associated with v is an infinitesimal invariant 8v that reflects the first-order variation of pairs (X-t, Z(t)). Over the years, 8v has been widely used in the study of various geometric questions. We note that whereas v is a transcendental invariant, like periods of algebraic integrals, delta(v) has a natural filtration whose associated graded gives algebraic sections of coherent sheaves. In a number of interesting cases, these sections have had geometric interpretations. In this paper, we will discuss an identification between the singularities v and delta(v). The formal proof of this result will be given in a separate work.
We discuss several structural properties of functions belonging to a parabolic energy class, reminiscent of the elliptic De Giorgi class. In earlier works, sub-potential lower bounds, giving insight into the structural behavior of elements of these classes, were established for the linear case: Here, we extend these results to the nonlinear one. By showing that sub-potential lower bounds follow solely from the Harnack inequality, we show that positive solutions to Trudinger's equation and elements of parabolic De Giorgi classes have a common lower bound. For both cases, we derive Liouville-type rigidity results in the parabolic setting.
Normal functions v provide a method for studying algebraic cycles Z(t) subset of X-t varying in a family of smooth projective varieties. Associated with v is an infinitesimal invariant 8v that reflects the first-order variation of pairs (X-t, Z(t)). Over the years, 8v has been widely used in the study of various geometric questions. We note that whereas v is a transcendental invariant, like periods of algebraic integrals, delta(v) has a natural filtration whose associated graded gives algebraic sections of coherent sheaves. In a number of interesting cases, these sections have had geometric interpretations. In this paper, we will discuss an identification between the singularities v and delta(v). The formal proof of this result will be given in a separate work.
The large sieve inequality, although best possible in general, is subject to improvements for some basic special sequences. We exemplify how the ideas of Bombieri and Davenport can be exploited to this end.
