Let (M,g) be a compact Riemannian manifold of dimension 3, and let \mathscr{F} denote the collection of all embedded surfaces homeomorphic to \mathbb{RP}^2. We study the infimum of the areas of all surfaces in \mathscr{F}. This quantity is related to the systole of (M,g). It makes sense whenever \mathscr{F} is non-empty.
In this paper, we give an upper bound for this quantity in terms of the minimum of the scalar curvature of (M,g). Moreover, we show that equality holds if and only if (M,g) is isometric to \mathbb{RP}^3 up to scaling.
%The proof uses the formula for the second variation of area, and Hamilton's Ricci flow.
Minimizing tangent maps into complex projective space
University of Chicago（芝加哥大学）；University of Cambridge（剑桥大学）；Instituto Superior Técnico（技术研究所）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学促进部数学研究所）
We prove a new inequality relating volume to length of closed geodesics on area minimizers for generic metrics on the complex projective plane. We exploit recent regularity results for area minimizers by Moore and White, and the Kronheimer--Mrowka proof of the Thom conjecture.
A Liouville classification of a natural Hamiltonian system on the projective plane with a rotation metric and a linear integral is obtained. All Fomenko--Zieschang invariants (i.e., labeled molecules) of the system are calculated.
A natural parametrization of smooth projective plane curves which tolerates the presence of sextactic points is the Forsyth-Laguerre parametrization. On a closed projective plane curve, which necessarily contains sextactic points, this parametrization is, however, in general not periodic. We show that by the introduction of an additional scalar parameter $α\leq \frac12$ one can define a projectively invariant $2π$-periodic global parametrization on every simple closed convex sufficiently smooth projective plane curve without inflection points. For non-quadratic curves this parametrization, whi…
The first paper in systolic geometry was published by Loewner's student P. M. Pu over half a century ago. Pu proved an inequality relating the systole and the area of an arbitrary metric on the real projective plane. We prove a stronger version of Pu's systolic inequality with a remainder term.
In this paper we compute the mapping class group of simply-connected closed smooth manifolds $M$ with integral homology $H_{*}(M) \cong \mathbb Z \oplus \mathbb Z \oplus \mathbb Z$ provided that $\dim M \ne 4$.
