Let $(M^4,g)$ be a complete four-dimensional Riemannian manifold. First, if the $Q$-curvature $Q_g\geq 6k^2$ and scalar curvature $R_g\geq -12k$ for some positive constant $k$, then $(M^4,g)$ is either Einstein with $Ric_g=-3kg$ or compact with $R_g\ge 12k$. As a corollary, the fundamental group $\pi_1(M^4)$ satisfies $|\pi_1(M^4)|\leq 16\pi^2/(\int_{M^4}Q_g d\mu_g),$ under the additional assumption $R_g>-12k$. Second, if the scalar curvature $R_g>0$ and $Q_g\geq \theta R_g$ for a positive constant $\theta$, then $M^4$ is compact and the diameter of $(M^4,g)$ is at most $4\pi/\sqrt{15\theta}$.
Let $n\geq2$ and let $M^n$ be a closed connected smooth manifold. Let $R^\gamma(M)$ be the space of smooth Riemannian metrics $g$ on $M$ for which the generalized conformal Laplace operator $-\gamma\Delta_g+\mathrm{R}_g$ is strictly positive. We prove that if $n=2$ and $\gamma>0$, or if $n\ge3$ and $0<\gamma \leq 4(n-1...
G. Antonelli, Georg Frenck, Bernhard Hanke· 0 citations
Let $I_a(g)=Q_g+a\sigma_2(A_g)$, where $A_g$ is the Schouten tensor and $Q_g$ is Branson's $Q$-curvature. On a closed connected manifold of dimension $n\ge4$ with a positive Einstein metric $g_0$, we prove that every smooth metric conformal to $g_0$ with nonnegative scalar curvature and constant $I_a(g)$ is Einstein fo...
Let $M^n$ be a closed oriented manifold of even dimension $n=2m\ge2$, equipped with a smooth metric $g_1$ and a Riemannian metric $g_2$. We compute the Wodzicki residue of $D_{g_1}^2D_{g_2}^{-n}$ on the common exterior bundle, where $D_{g_r}=d+\delta_{g_r}$. This is the noncommutative integral of $D_{g_1}^2$ relative t...
Let $(M^3,g)$ be a complete Riemannian manifold diffeomorphic to $\R^3\setminus\{0\}$, with nonnegative scalar curvature. Assume that a distinguished end is asymptotically flat, with ADM mass $m_+$. For each $p\in(1,3)$, define $c_{O,p}$ as the infimum of the Schwarzschild-normalized $p$-capacity over outward-minimizin...
Let $(M^n,g)$ be a smooth compact Riemannian manifold of dimension $n\ge 5$. We show that the existence of a conformal metric with positive $Q$-curvature $Q_g$ and positive scalar curvature $R_g$ is equivalent to the positivity of both the Yamabe invariant $Y(M^n,[g])$ and the Paneitz operator $P_g$. For $n=5$, this eq...
We prove local rigidity of the Euclidean metric and close conformally Euclidean ones for the anisotropic Calder\'on problem on smooth compact domains $M\subset\mathbb{R}^n$, $n\ge3$. A smooth Riemannian metric $g$ with the same induced boundary metric and Dirichlet-to-Neumann map as the background $g_0$ is isometric to...
P. Stefanov· 1 citation
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
