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Register a new userQ-curvature flow with indefinite nonlinearity
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Li Ma
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In this note, we study Q-curvature flow on $S^4$ with indefinite nonlinearity. Our result is that the prescribed Q-curvature problem on $S^4$ has a solution provided the prescribed Q-curvature $f$ has its positive part, which possesses non-degenerate critical points such that $Delta_{S^4} f ot=0$ at the saddle points and an extra condition such as a nontrivial degree counting condition.
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In this paper, we consider the indefinite scalar curvature problem on $R^n$. We propose new conditions on the prescribing scalar curvature function such that the scalar curvature problem on $R^n$ (similarly, on $S^n$) has at least one solution. The key observation in our proof is that we use the bifurcation method to get a large solution and then after establishing the Harnack inequality for solutions near the critical points of the prescribed scalar curvature and taking limit, we find the nontrivial positive solution to the indefinite scalar curvature problem.
We continue studying a parabolic flow of almost K{a}hler structures introduced by Streets and Tian which naturally extends K{a}hler-Ricci flow onto symplectic manifolds. In the system of primarily the symplectic form, almost complex structure, Chern torsion and Chern connection, we establish new formulas for the evolutions of canonical quantities, in particular those related to the Chern connection. Using this, we give an extended characterization of fixed points of the flow originally performed by Streets and Tian.
In this note, we study the curvature flow to Nirenberg problem on $S^2$ with non-negative nonlinearity. This flow was introduced by Brendle and Struwe. Our result is that the Nirenberg problems has a solution provided the prescribed non-negative Gaussian curvature $f$ has its positive part, which possesses non-degenerate critical points such that $Delta_{S^2} f>0$ at the saddle points.
We prove the mean curvature flow of a spacelike graph in $(Sigma_1times Sigma_2, g_1-g_2)$ of a map $f:Sigma_1to Sigma_2$ from a closed Riemannian manifold $(Sigma_1,g_1)$ with $Ricci_1> 0$ to a complete Riemannian manifold $(Sigma_2,g_2)$ with bounded curvature tensor and derivatives, and with sectional curvatures satisfying $K_2leq K_1$, remains a spacelike graph, exists for all time, and converges to a slice at infinity. We also show, with no need of the assumption $K_2leq K_1$, that if $K_1>0$, or if $Ricci_1>0$ and $K_2leq -c$, $c>0$ constant, any map $f:Sigma_1to Sigma_2$ is trivially homotopic provided $f^*g_2<rho g_1$ where $rho=min_{Sigma_1}K_1/sup_{Sigma_2}K_2^+geq 0$, in case $K_1>0$, and $rho=+infty$ in case $K_2leq 0$. This largely extends some known results for $K_i$ constant and $Sigma_2$ compact, obtained using the Riemannian structure of $Sigma_1times Sigma_2$, and also shows how regularity theory on the mean curvature flow is simpler and more natural in pseudo-Riemannian setting then in the Riemannian one.
In this paper, we employ a nonlocal $Q$-curvature flow inspired by Gursky-Malchiodis work cite{gur_mal} to solve the prescribed $Q$-curvature problem on a class of closed manifolds: For $n geq 5$, let $(M^n,g_0)$ be a smooth closed manifold, which is not conformally diffeomorphic to the standard sphere, satisfying either Gursky-Malchiodis semipositivity hypotheses: scalar curvature $R_{g_0}>0$ and $Q_{g_0} geq 0$ not identically zero or Hang-Yangs: Yamabe constant $Y(g_0)>0$, Paneitz-Sobolev constant $q(g_0)>0$ and $Q_{g_0} geq 0$ not identically zero. Let $f$ be a smooth positive function on $M^n$ and $x_0$ be some maximum point of $f$. Suppose either (a) $n=5,6,7$ or $(M^n,g_0)$ is locally conformally flat; or (b) $n geq 8$, Weyl tensor at $x_0$ is nonzero. In addition, assume all partial derivatives of $f$ vanish at $x_0$ up to order $n-4$, then there exists a conformal metric $g$ of $g_0$ with its $Q$-curvature $Q_g$ equal to $f$. This result generalizes Escobar-Schoens work [Invent. Math. 1986] on prescribed scalar curvature problem on any locally conformally flat manifolds of positive scalar curvature.
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