{
  "id": "2105.12625",
  "version": "v1",
  "published": "2021-05-26T15:28:28.000Z",
  "updated": "2021-05-26T15:28:28.000Z",
  "title": "Weighted monotonicity theorems and applications to minimal surfaces in hyperbolic space",
  "authors": [
    "Manh Tien Nguyen"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "We show that there is a weighted monotonicity theorem corresponding to each function on a Riemannian manifold whose Hessian is a multiple of the metric tensor. Such functions appear in Euclidean space, hyperbolic space ( \\mathbb{H}^n ) and the round sphere ( \\mathbb{S}^n ) as the distance function, the Minkowskian coordinates of ( \\mathbb{R}^{n,1} ) and the Euclidean coordinates of ( \\mathbb{R}^{n+1} ). In ( \\mathbb{H}^n ), the time-weighted monotonicity theorem is compared with the unweighted version due to Anderson in \\cite{Anderson82_CompleteMinimalVarieties}. Applications include upper bounds for Graham--Witten renormalised area of minimal surfaces in term of the length of boundary curve and a complete computation of Alexakis--Mazzeo degrees defined in \\cite{Alexakis.Mazzeo10_RenormalizedAreaProperly}. An argument on area-minimising cones suggests existence of a minimal surface in ( \\mathbb{H}^4 ) bounded by the Hopf link ( \\{(z,w)\\in \\mathbb{C}^2: zw=\\epsilon > 0, |z|^2 +|w|^2 = 1\\} ) other than the pair of disks. We give an explicit construction of a minimal annulus in ( \\mathbb{H}^4 ) with this property. We also obtain its sister in ( \\mathbb{S}^4 ) by the same method. A weighted monotonicity theorem is also proved in Riemannian manifolds whose sectional curvature is bounded from above.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-26T15:28:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53A10",
      "53C43"
    ],
    "keywords": [
      "minimal surface",
      "hyperbolic space",
      "applications",
      "riemannian manifold",
      "functions appear"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}