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Register a new userThe Prescribed $Q$-Curvature Flow for Arbitrary Even Dimension in a Critical Case
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In this paper, we study the prescribed $Q$-curvature flow equation on a arbitrary even dimensional closed Riemannian manifold $(M,g)$, which was introduced by S. Brendle in cite{B2003}, where he proved the flow exists for long time and converges at infinity if the GJMS operator is weakly positive with trivial kernel and $int_M Qdmu < (n-1)!Volleft( S^n right) $. In this paper we study the critical case that $int_M Qdmu = (n-1)!Volleft( S^n right)$, we will prove the convergence of the flow under some geometric hypothesis. In particular, this gives a new proof of Li-Li-Lius existence result in cite{LLL2012} in dimensiona 4 and extend the work of Li-Zhu cite{LZ2019} in dimension 2 to general even dimensions. In the proof, we give a explicit expression of the limit of the corresponding energy functional when the blow up occurs.
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On Riemannian manifolds of dimension 4, for prescribed scalar curvature equation, under lipschitzian condition on the prescribed curvature, we have an uniform estimate for the solutions of the equation if we control their minimas.
We consider the heat equation associated with a class of hypoelliptic operators of Kolmogorov-Fokker-Planck type in dimension two. We explicitly compute the first meaningful coefficient of the small time asymptotic expansion of the heat kernel on the diagonal, and we interpret it in terms of curvature-like invariants of the optimal control problem associated with the diffusion. This gives a first example of geometric interpretation of the small-time heat kernel asymptotics of non-homogeneous Hormander operators which are not associated with a sub-Riemannian structure, i.e., whose second-order part does not satisfy the Hormander condition.
We will give a new proof of the existence of hypercylinder expander of the inverse mean curvature flow which is a radially symmetric homothetic soliton of the inverse mean curvature flow in $mathbb{R}^ntimes mathbb{R}$, $nge 2$, of the form $(r,y(r))$ or $(r(y),y)$ where $r=|x|$, $xinmathbb{R}^n$, is the radially symmetric coordinate and $yin mathbb{R}$. More precisely for any $lambda>frac{1}{n-1}$ and $mu>0$, we will give a new proof of the existence of a unique even solution $r(y)$ of the equation $frac{r(y)}{1+r(y)^2}=frac{n-1}{r(y)}-frac{1+r(y)^2}{lambda(r(y)-yr(y))}$ in $mathbb{R}$ which satisfies $r(0)=mu$, $r(0)=0$ and $r(y)>yr(y)>0$ for any $yinmathbb{R}$. We will prove that $lim_{ytoinfty}r(y)=infty$ and $a_1:=lim_{ytoinfty}r(y)$ exists with $0le a_1<infty$. We will also give a new proof of the existence of a constant $y_1>0$ such that $r(y_1)=0$, $r(y)>0$ for any $0<y<y_1$ and $r(y)<0$ for any $y>y_1$.
We consider an evolution problem associated to the Kazdan-Warner equation on a closed Riemann surface $(Sigma,g)$ begin{align*} -Delta_{g}u=8pileft(frac{he^{u}}{int_{Sigma}he^{u}{rm d}mu_{g}}-frac{1}{int_{Sigma}{rm d}mu_{g}}right) end{align*} where the prescribed function $hgeq0$ and $max_{Sigma}h>0$. We prove the global existence and convergence under additional assumptions such as begin{align*} Delta_{g}ln h(p_0)+8pi-2K(p_0)>0 end{align*} for any maximum point $p_0$ of the sum of $2ln h$ and the regular part of the Green function, where $K$ is the Gaussian curvature of $Sigma$. In particular, this gives a new proof of the existence result by Yang and Zhu [Proc. Amer. Math. Soc. 145 (2017), no. 9, 3953-3959] which generalizes existence result of Ding, Jost, Li and Wang [Asian J. Math. 1 (1997), no. 2, 230-248] to the non-negative prescribed function case.
We will give a new proof of a recent result of P.~Daskalopoulos, G.Huisken and J.R.King ([DH] and reference [7] of [DH]) on the existence of self-similar solution of the inverse mean curvature flow which is the graph of a radially symmetric solution in $mathbb{R}^n$, $nge 2$, of the form $u(x,t)=e^{lambda t}f(e^{-lambda t} x)$ for any constants $lambda>frac{1}{n-1}$ and $mu<0$ such that $f(0)=mu$. More precisely we will give a new proof of the existence of a unique radially symmetric solution $f$ of the equation $mbox{div},left(frac{ abla f}{sqrt{1+| abla f|^2}} right)=frac{1}{lambda}cdotfrac{sqrt{1+| abla f|^2}}{xcdot abla f-f}$ in $mathbb{R}^n$, $f(0)=mu$, for any $lambda>frac{1}{n-1}$ and $mu<0$, which satisfies $f_r(r)>0$, $f_{rr}(r)>0$ and $rf_r(r)>f(r)$ for all $r>0$. We will also prove that $lim_{rtoinfty}frac{rf_r(r)}{f(r)}=frac{lambda (n-1)}{lambda (n-1)-1}$.
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