An Upper Bound of the Total Q-Curvature and Its Isoperimetric Deficit for Higher-dimensional Conformal Euclidean Metrics

The aim of this paper is to give not only an explicit upper bound of the total Q-curvature but also an induced isoperimetric deficit formula for the complete conformal metrics on $mathbb R^n$, $nge 3$ with scalar curvature being nonnegative near infinity and Q-curvature being absolutely convergent.

Optimal Monotonicity of $L^p$ Integral of Conformal Invariant Green Function

Both analytic and geometric forms of an optimal monotone principle for $L^p$-integral of the Green function of a simply-connected planar domain $Omega$ with rectifiable simple curve as boundary are established through a sharp one-dimensional power integral estimate of Riemann-Stieltjes type and the Huber analytic and geometric isoperimetric inequalities under finiteness of the positive part of total Gauss curvature of a conformal metric on $Omega$. Consequently, new analytic and geometric isoperimetric-type inequalities are discovered. Furthermore, when applying the geometric principle to two-dimensional Riemannian manifolds, we find fortunately that ${0,1}$-form of the induced principle is midway between Moser-Trudingers inequality and Nash-Sobolevs inequality on complete noncompact boundary-free surfaces, and yet equivalent to Nash-Sobolevs/Faber-Krahns eigenvalue/Heat-kernel-upper-bound/Log-Sobolevs inequality on the surfaces with finite total Gauss curvature and quadratic area growth.

Toward Best Isoperimetric Constants for $(H^1,BMO)$-Normal Conformal Metrics on $mathbb R^n$, $nge 3$

The aim of this article is: (a) To establish the existence of the best isoperimetric constants for the $(H^1,BMO)$-normal conformal metrics $e^{2u}|dx|^2$ on $mathbb R^n$, $nge 3$, i.e., the conformal metrics with the Q-curvature orientated conditions $$ (-Delta)^{n/2}uin H^1(mathbb R^n) & u(x)=hbox{const.}+frac{int_{mathbb R^n}(logfrac{|cdot|}{|x-cdot|})(-Delta)^{n/2} u(cdot) dmathcal{H}^n(cdot)}{2^{n-1}pi^{n/2}Gamma(n/2)}; $$ (b) To prove that $(nomega_n^frac1n)^frac{n}{n-1}$ is the optimal upper bound of the best isoperimetric constants for the complete $(H^1,BMO)$-normal conformal metrics with nonnegative scalar curvature; (c) To find the optimal upper bound of the best isoperimetric constants via the quotients of two power integrals of Greens functions for the $n$-Laplacian operators $-hbox{div}(| abla u|^{n-2} abla u)$.

Greens formula with $bbc^{*}$-action and Caldero-Kellers formula for cluster algebras

It is known that Greens formula over finite fields gives rise to the comultiplications of Ringel-Hall algebras and quantum groups (seecite{Green}, also see cite{Lusztig}). In this paper, we deduce the projective version of Greens formula in a geometric way. Then following the method of Hubery in cite{Hubery2005}, we apply this formula to proving Caldero-Kellers multiplication formula for acyclic cluster algebras of arbitrary type.

The $mathcal{Q}_p$ Carleson Measure Problem

Let $mu$ be a nonnegative Borel measure on the open unit disk $mathbb{D}subsetmathbb{C}$. This note shows how to decide that the Mobius invariant space $mathcal{Q}_p$, covering $mathcal{BMOA}$ and $mathcal{B}$, is boundedly (resp., compactly) embedded in the quadratic tent-type space $T^infty_p(mu)$. Interestingly, the embedding result can be used to determine the boundedness (resp., the compactness) of the Volterra-type and multiplication operators on $mathcal{Q}_p$.