Add to ORKGFor a closed immersed minimal submanifold $M^n$ in the unit sphere $\mathbb{S}^{N}$ $(n<N)$, we prove $$ {\rm Vol}(M^n) \geq \frac{m}{2}{\rm Vol}(\mathbb{S}^{n})+m\frac{\sqrt{n+1}}{n} {\rm Vol}(\mathbb{S}^{n-1}),$$ where $m$ denotes the maximal multiplicity of intersection points of $M^n$ in $ \mathbb{S}^{N}$ and ${\rm Vol}$ denotes the Riemannian volume functional. As an application, if the volume of $M^n$ is less than or equal to the volume of any $n$-dimensional minimal Clifford torus, then $M^n$ must be embedded, verifying the non-embedded case of Yau's conjecture. In addition, we also get volume gaps for hypersurfaces under some conditions.Comment: 15 pages, any comments are welcom
AbstractWe improve the well-known scalar curvature pinching theorem due to Peng–Terng for n (n⩽5)-di...
Given $M$ a Riemannian manifold with (possibly empty) boundary, we show that its volume spectrum $\{...
In the early 1980s, S. T. Yau conjectured that any compact Riemannian three-manifold admits an infin...
summary:For closed immersed submanifolds of Euclidean spaces, we prove that $\int |\mu |^2\, dV\geq ...
Given an $n$-dimensional Riemannian sphere conformal to the round one and $\delta$-pinched, we show ...
We consider the problem of minimal volume vector fields on a given Riemann surface, specialising on ...
We prove that in a Riemannian manifold $M$, each function whose Hessian is proportional the metric t...
We prove a sharp area estimate for minimal submanifolds that pass through a prescribed point in a ge...
We discover new monotonicity formulae for minimal submanifolds in space forms, which imply the sharp...
We prove that the four-dimensional round sphere contains a minimally embedded hypertorus, as well as...
In Guaraco (J. Differential Geom. 108(1):91–133, 2018) a new proof was given of the existence of a c...
We prove that every complete non-compact manifold of finite volume contains a (possibly non-compact)...
AbstractWe use minimal Legendrian submanifolds in spheres to construct examples of absolutely area-m...
In this dissertation we develop new variational methods with the aim to build minimal k-submanifolds...
In this paper, we prove that in any compact Riemannian manifold with smooth boundary, of dimension a...
AbstractWe improve the well-known scalar curvature pinching theorem due to Peng–Terng for n (n⩽5)-di...
Given $M$ a Riemannian manifold with (possibly empty) boundary, we show that its volume spectrum $\{...
In the early 1980s, S. T. Yau conjectured that any compact Riemannian three-manifold admits an infin...
summary:For closed immersed submanifolds of Euclidean spaces, we prove that $\int |\mu |^2\, dV\geq ...
Given an $n$-dimensional Riemannian sphere conformal to the round one and $\delta$-pinched, we show ...
We consider the problem of minimal volume vector fields on a given Riemann surface, specialising on ...
We prove that in a Riemannian manifold $M$, each function whose Hessian is proportional the metric t...
We prove a sharp area estimate for minimal submanifolds that pass through a prescribed point in a ge...
We discover new monotonicity formulae for minimal submanifolds in space forms, which imply the sharp...
We prove that the four-dimensional round sphere contains a minimally embedded hypertorus, as well as...
In Guaraco (J. Differential Geom. 108(1):91–133, 2018) a new proof was given of the existence of a c...
We prove that every complete non-compact manifold of finite volume contains a (possibly non-compact)...
AbstractWe use minimal Legendrian submanifolds in spheres to construct examples of absolutely area-m...
In this dissertation we develop new variational methods with the aim to build minimal k-submanifolds...
In this paper, we prove that in any compact Riemannian manifold with smooth boundary, of dimension a...
AbstractWe improve the well-known scalar curvature pinching theorem due to Peng–Terng for n (n⩽5)-di...
Given $M$ a Riemannian manifold with (possibly empty) boundary, we show that its volume spectrum $\{...
In the early 1980s, S. T. Yau conjectured that any compact Riemannian three-manifold admits an infin...