Determining a Riemannian Metric from Minimal Areas

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We prove that if (M,g) is a topological 3-ball with a C4-smooth Riemannian metric g, and mean-convex boundary ∂M then knowledge of least areas circumscribed by simple closed curves γ⊂∂M uniquely determines the metric g, under some additional geometric assumptions. These are that g is either a) C3-close to Euclidean or b) satisfies much weaker geometric conditions which hold when the manifold is to a sufficient degree either thin, or straight. %sufficiently thin. In fact, the least area data that we require is for a much more restricted class of curves γ⊂∂M. We also prove a corresponding local result: assuming only that (M,g) has strictly mean convex boundary at a point p∈∂M, we prove that knowledge of the least areas circumscribed by any simple closed curve γ in a neighbourhood U⊂∂M of p uniquely determines the metric near p. Additionally, we sketch the proof of a global result with no thin/straight or curvature condition, but assuming the metric admits minimal foliations "from all directions". The proofs rely on finding the metric along a continuous sweep-out of M by area-minimizing surfaces; they bring together ideas from the 2D-Calderón inverse problem, minimal surface theory, and the careful analysis of a system of pseudo-differential equations.
Originalspråkengelska
Utgivningsår2017
StatusInsänt - 2017
MoE-publikationstypEj behörig

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Balehowsky, T. (2017). Determining a Riemannian Metric from Minimal Areas. Manuskript har skickats in för publicering.
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title = "Determining a Riemannian Metric from Minimal Areas",
abstract = "We prove that if (M,g) is a topological 3-ball with a C4-smooth Riemannian metric g, and mean-convex boundary ∂M then knowledge of least areas circumscribed by simple closed curves γ⊂∂M uniquely determines the metric g, under some additional geometric assumptions. These are that g is either a) C3-close to Euclidean or b) satisfies much weaker geometric conditions which hold when the manifold is to a sufficient degree either thin, or straight. {\%}sufficiently thin. In fact, the least area data that we require is for a much more restricted class of curves γ⊂∂M. We also prove a corresponding local result: assuming only that (M,g) has strictly mean convex boundary at a point p∈∂M, we prove that knowledge of the least areas circumscribed by any simple closed curve γ in a neighbourhood U⊂∂M of p uniquely determines the metric near p. Additionally, we sketch the proof of a global result with no thin/straight or curvature condition, but assuming the metric admits minimal foliations {"}from all directions{"}. The proofs rely on finding the metric along a continuous sweep-out of M by area-minimizing surfaces; they bring together ideas from the 2D-Calder{\'o}n inverse problem, minimal surface theory, and the careful analysis of a system of pseudo-differential equations.",
author = "Tracey Balehowsky",
year = "2017",
language = "English",
type = "Other",

}

Determining a Riemannian Metric from Minimal Areas. / Balehowsky, Tracey.

2017, .

Forskningsoutput: Övriga bidragForskningPeer review

TY - GEN

T1 - Determining a Riemannian Metric from Minimal Areas

AU - Balehowsky, Tracey

PY - 2017

Y1 - 2017

N2 - We prove that if (M,g) is a topological 3-ball with a C4-smooth Riemannian metric g, and mean-convex boundary ∂M then knowledge of least areas circumscribed by simple closed curves γ⊂∂M uniquely determines the metric g, under some additional geometric assumptions. These are that g is either a) C3-close to Euclidean or b) satisfies much weaker geometric conditions which hold when the manifold is to a sufficient degree either thin, or straight. %sufficiently thin. In fact, the least area data that we require is for a much more restricted class of curves γ⊂∂M. We also prove a corresponding local result: assuming only that (M,g) has strictly mean convex boundary at a point p∈∂M, we prove that knowledge of the least areas circumscribed by any simple closed curve γ in a neighbourhood U⊂∂M of p uniquely determines the metric near p. Additionally, we sketch the proof of a global result with no thin/straight or curvature condition, but assuming the metric admits minimal foliations "from all directions". The proofs rely on finding the metric along a continuous sweep-out of M by area-minimizing surfaces; they bring together ideas from the 2D-Calderón inverse problem, minimal surface theory, and the careful analysis of a system of pseudo-differential equations.

AB - We prove that if (M,g) is a topological 3-ball with a C4-smooth Riemannian metric g, and mean-convex boundary ∂M then knowledge of least areas circumscribed by simple closed curves γ⊂∂M uniquely determines the metric g, under some additional geometric assumptions. These are that g is either a) C3-close to Euclidean or b) satisfies much weaker geometric conditions which hold when the manifold is to a sufficient degree either thin, or straight. %sufficiently thin. In fact, the least area data that we require is for a much more restricted class of curves γ⊂∂M. We also prove a corresponding local result: assuming only that (M,g) has strictly mean convex boundary at a point p∈∂M, we prove that knowledge of the least areas circumscribed by any simple closed curve γ in a neighbourhood U⊂∂M of p uniquely determines the metric near p. Additionally, we sketch the proof of a global result with no thin/straight or curvature condition, but assuming the metric admits minimal foliations "from all directions". The proofs rely on finding the metric along a continuous sweep-out of M by area-minimizing surfaces; they bring together ideas from the 2D-Calderón inverse problem, minimal surface theory, and the careful analysis of a system of pseudo-differential equations.

UR - https://arxiv.org/abs/1711.09379

M3 - Other contribution

ER -