Hypersurfaces in hyperbolic space with support function

Abstract

a b s t r a c tProstaglandin F2a (PGF) treatment is routinely used in the reproductive management of mares to induce luteolysis and allow a subsequent return to estrus. The objective of this retrospective study was to assess the effect of follicle size at the time of administration of cloprostenol on interval to subsequent ovulation. A secondary objective was to determine the incidence of hemorrhagic anovulatory follicle (HAF) formation after PGF administration. Reproductive records of 275 mares monitored over a total of 520 estrous cycles were evaluated. All mares received a single intramuscular dose of 250 mg of the synthetic PGF analog cloprostenol sodium between days 5 and 12 after ovulation. The average interval from PGF to ovulation was 8.4 AE 2.5 days. The interval from PGF administration to subsequent ovulation was inversely proportional to the diameter of the largest follicle at the time of treatment. Administration of cloprostenol to mares with a large (!35 mm in diameter) diestrous follicle resulted in one of three outcomesdovulation within 48 hours (13.4%) with variable uterine edema, ovulation after 48 hours usually accompanied by the presence of uterine edema (73.1%), or regression without ovulation followed by emergence and eventual ovulation of a new dominant follicle (13.4%). There was no effect of mare age or season on interval from PGF to ovulation. The overall incidence of HAF development after PGF administration in this study was low (2.5%).

Min‐Oo Conjecture for Fully Nonlinear Conformally Invariant Equations

“…R nC1 is an embedding for all 2 OE0; 0 /. Thus, summarizing all we have done in this subsection, we obtain the following: 2,8]). Let S n be a relatively compact domain with smooth boundary and 2 C 2 .…”

“…Here we recover the hypersurface interpretation of conformal metrics on the sphere developed in [2,8]. Let us denote by L nC2 the Minkowski spacetime, that is, the vector space R nC2 endowed with the Minkowski spacetime metric h ; i given by…”

“…Thus, @D.p i ; i / \ ƒ is empty for every i , and, in view of property (ii) in [15, theorem 1.4], the balls x D.p i ; i / are mutually disjoint. Let us turn to point (2). If ƒ is empty, the collection fD.p i ; i /g of balls must be finite thanks to property (iii) in [15, theorem 1.4], in which case the simple connectedness of G is clear.…”

“…PROOF. Note that, since x 2 @S n C , t .x/ 2 H.x; t /, where H.x; t / is the horosphere whose point at infinity is x and signed distance to the origin is t > 0 (see [2]). This proves the first part of the claim.…”

“…Our approach relies in a geometric method developed by the third author, Gálvez and Mira in [8], and further developments contained in [1][2][3]6,7], where conformal metrics on spherical domains are represented by hypersurfaces in the hyperbolic space. In order to reduce our problem on locally conformally flat manifolds to conformal metrics on subdomains of the sphere, we use results contained in the work of Spiegel [22] and Li and Nguyen [15] based on the deep theory by Schoen and Yau [20] on the developing map of a locally conformally flat manifold.…”