arXiv · math/0507199
See math.CV/0509030 which replaces this paper.
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A. V. Isaev. 2006-11-16. Hyperbolic 2-dimensional manifolds with 3-dimensional automorphism groups. https://arxiv.org/abs/math/0507199
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A plurisubharmonic weight is \emph{log canonical} if it is at the critical point of turning non-integrable. Given a log canonical plurisubharmonic weight, we show that locally there always exists a log canonical `holomorphic' weight having the same non-integrable locus. The proof uses the theory of quasimonomial valuations developed by Jonsson-Mustaţă and Xu, as well as the minimal model program over complex analytic spaces due to Fujino and Lyu-Murayama. As a consequence, we obtain that the non-integrable locus is seminormal.
Each homeomorphic parametrization of a Jordan curve via the unit circle $\mathbb{S}\subset \mathbb{S}^2$ extends to a homeomorphism of the entire $\mathbb{S}^2.$ We show that there exists a Jordan curve of positive area whose every parametrization via $\mathbb{S}$ admits a homeomorphic extension that has $p$-Sobolev regularity on $\mathbb{S}^2$ for all exponents $1\le p<2 .$ This is obtained via a new sharp extension result for the case of $p<2.$
We disprove a conjecture of Ruscheweyh, Salinas and Sugawa concerning the universal convexity of the normalized generating function associated with the Ramanujan sequence $θ(n).$ More precisely, we prove that the function \[ σ(z)=\frac{1}{θ(1)-1/3}\sum_{n\geq1}(θ(n)-1/3)z^{n} \] is not universally convex on the slit domain $\mathbb{C}\backslash[1,\infty).$ The proof uses Watson\textquoteright s integral representation of $θ(n),$ asymptotic estimates for an auxiliary function $h,$ and the representation of derivatives of universally convex functions via logarithmic integrals. As a general extension of this result, we also show that the generalized Ramanujan generating function \[ σ_α(z)=\frac{1}{θ(1)-1/3}\sum_{n\geq1}(θ(n)-1/3)\frac{z^{n}}{n^α} \] is universally convex if and only if $α\geq1.$