Ailana Fraser


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Graduate Student Supervision

Doctoral Student Supervision (Jan 2008 - Nov 2019)
Index bounds and existence results for minimal surfaces and harmonic maps (2018)

In this work, we focus on three problems. First, we give a relationship between the eigenvalues of the Hodge Laplacian and the eigenvalues of the Jacobi operator for a free boundary minimal hypersurface of a Euclidean convex body. We then use this relationship to obtain new index bounds for such minimal hypersurfaces in terms of their topology. In particular, we show that the index of a free boundary minimal surface in a convex domain in ℝ³ tends to infinity as its genus or the number of boundary components tends to infinity. Second, we consider the relationship between the kth normalized eigenvalue of the Dirichlet-to-Neumann map (the kth Steklov eigenvalue) and the geometry of rotationally symmetric Möbius bands. More specifically, we look at the problem of finding a metric that maximizes the kth Steklov eigenvalue among all rotationally symmetric metrics on the Möbius band. We show that such a metric can always be found and that it is realized by the induced metric on a free boundary minimal Möbius band in B⁴. Third, we consider the existence problem for harmonic maps into CAT(1) spaces. If Σ is a compact Riemann surface, X is a compact locally CAT(1) space and φ : Σ → X is a continuous finite energy map, we use the technique of harmonic replacement to prove that either there exists a harmonic map u : Σ → X homotopic to φ or there exists a conformal harmonic map v : S² → X. To complete the argument, we prove compactness for energy minimizers and a removable singularity theorem for conformal harmonic maps.

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Uniqueness of Lagrangian mean curvature flow and minimal immersions with free boundary (2013)

In this thesis we investigate some problems on the uniqueness of mean curvatureflow and the existence of minimal surfaces, by geometric and analyticmethods. A summary of the main results is as follows.(i) The special Lagrangian submanifolds form a very important class ofminimal submanifolds, which can be constructed via the method ofmean curvature flow. In the graphical setting, the potential functionfor the Lagrangian mean curvature ow satisfies a fully nonlinearparabolic equation [formula omitted]where the ⋋j's are the eigenvalues of the Hessian D²u.We prove a uniqueness result for unbounded solutions of (1) withoutany growth condition, via the method of viscosity solutions ([4], [13]):for any continuous u₀ in ℝn, there is a unique continuous viscositysolution to (1) in ℝn x [0;∞).(ii) Let N be a complete, homogeneously regular Riemannian manifoldof dimN ≥ 3 and let M be a compact submanifold of N. Let Ʃ bea compact Riemann surface with boundary. A branched immersionu : (Σ,∂Σ) → (N,M) is a minimal surface with free boundary in Mif u(Σ) has zero mean curvature and u(Σ) is orthogonal to M alongu(∂Σ)⊑ M.We study the free boundary problem for minimal immersions of compactbordered Riemann surfaces and prove that Σ if is not a disk, then there exists a free boundary minimalimmersion of Σ minimizing area in any given conjugacy class ofa map in C⁰(Σ,∂Σ;N,M) that is incompressible; the kernel of i* : π₁(M) → π₁(N) admits a generating set suchthat each member is freely homotopic to the boundary of an areaminimizing disk that solves the free boundary problem. (iii) Under certain nonnegativity assumptions on the curvature of a 3-manifold N and convexity assumptions on the boundary M=∂N, we investigate controlling topology for free boundary minimal surfaces of low index:• We derive bounds on the genus, number of boundary components;• We prove a rigidity result;• We give area estimates in term of the scalar curvature of N.

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Master's Student Supervision (2010 - 2018)
Regularity of Minimal Surfaces: A Self-contained Proof (2015)

In this thesis, a self-contained proof is given of the regularity of minimal surfaces via viscositysolutions, following the ideas of L.Caffarelli,X.Cabré [2], O.Savin[11][12], E.Giusti[7] and J.Roquejoffre[8], where we expand upon the ideas and give full details on the approach. Basically the proof of the program consists of four parts: 1) Density and measure estimates, 2) Viscosity solution methods of elliptic equations , 3) a geometric Harnack inequality and 4) iteration of the De Giorgi flatness result.

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Minimal hypersurfaces of the round sphere (2013)

The purpose of this thesis is to discuss a conjectured classification concerning the index of non-totally geodesic minimal hypersurfaces of the n-dimensional standard sphere of radius one S^n. We briefly discuss the basic theory of minimal submanifolds before turning our attention to minimal submanifolds and hypersurfaces in S^n. We present some results of Simons which show that any minimal submanifold of S^n is unstable, and how the totally geodesic S^k ⊂ S^n are characterized by their index. We then present a related conjecture which claims that the Clifford hypersurfaces are also characterized by their index in a similar way, discuss the most recent developments related to the conjecture, and give Urbano’s proof of the conjecture for the special case when n = 3

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Inradius Bounds for Stable, Minimal Surfaces in 3-Manifolds with Positive Scalar Curvature (2012)

Concrete topological properties of a manifold can be found by examining its geometry. Theorem 17 of his thesis, due to Myers [Mye41], is one such example of this; it gives an upper bound on the length of any minimizing geodesic in a manifold N in terms of a lower positive bound on the Ricci curvature of N, and concludes that N is compact. Our main result, Theorem 40, is of the same flavour as this, but we are instead concerned with stable, minimal surfaces in manifolds of positive scalar curvature. This result is a version of Proposition 1 in the paper of Schoen and Yau [SY83], written in the context of Riemannian geometry. It states: a stable, minimal 2-submanifold of a 3-manifold whose scalar curvature is bounded below by κ > 0 has a inradius bound of ≤√(8/3) π/√κ, and in particular is compact.

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The isoperimetric inequality in R^n (2011)

This thesis presents a complete proof of the isoperimetric inequality for a smooth surface in Euclidean space. The proof uses the Brunn-MinkowskiInequality, the formulae for the first variations of area and Alexandrov’s theorem.

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