A Sharp Estimate for the Genus of Embedded Surfaces in the 3-Sphere

Abstract

AbstractBy refining the volume estimate of Heintze and Karcher [11], we obtain a sharp pinching estimate for the genus of a surface in $$\mathbb S^{3}$$ S 3 , which involves an integral of the norm of its traceless second fundamental form. More specifically, we show that if g is the genus of a closed orientable surface $$\Sigma $$ Σ in a 3-dimensional orientable Riemannian manifold M whose sectional curvature is bounded below by 1, then $$4 \pi ^{2} g(\Sigma ) \le 2\left( 2 \pi ^{2}-|M|\right) +\int _{\Sigma } f(|{\mathop {A}\limits ^{\circ }}|)$$ 4 π 2 g ( Σ ) ≤ 2 2 π 2 - | M | + ∫ Σ f ( | A ∘ | ) , where $$ {\mathop {A}\limits ^{\circ }} $$ A ∘ is the traceless second fundamental form and f is an explicit function. As a result, the space of closed orientable embedded minimal surfaces $$\Sigma $$ Σ with uniformly bounded $$\Vert A\Vert _{L^3(\Sigma )}$$ ‖ A ‖ L 3 ( Σ ) is compact in the $$C^k$$ C k topology for any $$k\ge 2$$ k ≥ 2 .