تقرير
Weighted monotonicity theorems and applications to minimal surfaces in $\mathbb{H}^n$ and $S^n$
| العنوان: | Weighted monotonicity theorems and applications to minimal surfaces in $\mathbb{H}^n$ and $S^n$ |
|---|---|
| المؤلفون: | Nguyen, Manh Tien |
| سنة النشر: | 2021 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Differential Geometry, 53A10, 53C43 |
| الوصف: | We prove that in a Riemannian manifold $M$, each function whose Hessian is proportional the metric tensor yields a weighted monotonicity theorem. Such function appears in the Euclidean space, the round sphere $S^n$ and the hyperbolic space $\mathbb{H}^n$ as the distance function, the Euclidean coordinates of $\mathbb{R}^{n+1}$ and the Minkowskian coordinates of $\mathbb{R}^{n,1}$. Then we show that weighted monotonicity theorems can be compared and that in the hyperbolic case, this comparison implies three $SO(n,1)$-distinct unweighted monotonicity theorems. From these, we obtain upper bounds of the Graham--Witten renormalised area of a minimal surface in term of its ideal perimeter measured under different metrics of the conformal infinity. Other applications include a vanishing result for knot invariants coming from counting minimal surfaces of $\mathbb{H}^n$ and a quantification of how antipodal a minimal submanifold of $S^n$ has to be in term of its volume. Comment: 19 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2105.12625 |
| رقم الأكسشن: | edsarx.2105.12625 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |