It is conjectured that a complete, orientable Einstein four-manifold with strictly positive sectional curvature must be isometric to either the round four-sphere $S^4$ or the complex projective plane $\mathbb{CP}^2$ with the Fubini--Study metric. In this paper, we confirm this conjecture when the manifold is homeomorphic to $S^4$ or $\mathbb{CP}^2$. More precisely, we show that a complete, orientable Einstein four-manifold $M$ with strictly positive sectional curvature and Euler characteristic $\chi(M) \le 3$ (equivalently, by Freedman's classification theorem, homeomorphic to $S^4$ or $\mathbb{CP}^2$) is isometric to either the round $S^4$ or the Fubini--Study $\mathbb{CP}^2$. As an application, we obtain that a complete, orientable Einstein four-manifold with strictly positive sectional curvature admitting a nontrivial Killing field is isometric to either the round $S^4$ or the Fubini--Study $\mathbb{CP}^2$.
We construct smooth Einstein metrics with positive scalar curvature on $\#_k({\mathbb S^2}\times {\mathbb S^2})$ and on $k\mathbb{CP}^2\#(2k+1)\overline{\mathbb{CP}}{}^{2}$, for $k\geqslant2$. In particular, we show that positive Einstein metrics exist on infinitely many homeomorphism types of closed $4$-manifolds. The...
Let $(M,g)$ be a compact oriented Einstein four-manifold with Einstein constant $E$ and let $\widehat{R}^+$ denote the action of the Riemann curvature tensor on self-dual two-forms. We show that if $\widehat{R}^+<0$, then $g$ is strictly linearly stable for the Einstein-Hilbert functional, thus giving a chiral criterio...
Let $M^3\to X^4$ be a complete, connected, two-sided stable minimal immersion. We prove that if the ambient sectional curvature is nonnegative and the ambient scalar curvature has a positive uniform lower bound, then $M$ is totally geodesic and its normal Ricci curvature vanishes. No weak bounded geometry assumption an...
Let $\Gamma<SO(4)$ be a nontrivial finite group acting freely on $S^3$, and let $M=S^3/\Gamma$. We prove that every smooth conformal class $C$ on $M$ satisfies \[ Y(M,C)\le |\Gamma|^{-2/3}Y(S^3), \] with equality for the quotient-round conformal class. Consequently, \[ \sigma(M)=|\Gamma|^{-2/3}Y(S^3). \] Our approach u...
We prove that the exotic smooth $8$-sphere and both oriented homotopy $10$-spheres representing elements of order three admit Riemannian metrics with strictly positive sectional curvature. The construction combines Speran\c{c}a's special $S^3$-$S^3$ bundle models with the compatible-disk construction of He, Liu and Yau...
Let $X$ be a closed, smooth, orientable, positive definite $4$-manifold with $H_1(X ; \mathbb{Z}) = 0$. Let $p$ be a prime and suppose that $G = \mathbb{Z}_p$ acts smoothly on $X$ (if $p=2$ we also require an assumption on how $G$ acts on $H^2(X ; \mathbb{Z})$). Using equivariant Yang--Mills theory, Hambleton--Lee and...
David Baraglia· 0 citations
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