# Ailana Fraser

#### Relevant Degree Programs

## Graduate Student Supervision

##### Doctoral Student Supervision (Jan 2008 - April 2022)

In this work, we focus on three problems. First, we give a relationship between the number of eigenvalues of the Jacobi operator below a certain threshold and the topology of closed constant mean curvature (CMC) surfaces in three-dimensional Riemannian manifolds. We then obtain that the (weak) Morse index of CMC surfaces in an arbitrary 3-manifold is bounded below by a linear function of the genus when the constant mean curvature is greater than a certain nonnegative value. In particular, this implies that stable CMC surfaces are topological spheres. Corresponding results for CMC surfaces with free boundary in 3-manifolds with boundary are obtained as well. Second, we consider the space of embedded free boundary CMC surfaces with bounded topology, bounded area, and bounded boundary length in a 3-manifold N with boundary. We show that this space is almost compact in the sense that any sequence of surfaces in this space has a convergent subsequence that converges to a free boundary CMC surface, graphically and smoothly except on a finite set of singularities. If in addition Ric_N>0 and the boundary of N is convex, then the convergence is at most 2-sheeted. In particular, it is 1-sheeted if the limiting surface is not a minimal surface. Third, we consider the maximization of Steklov eigenvalues in higher dimensions. We show that for compact manifolds of dimension at least 3 with nonempty boundary, we can modify the manifold by performing surgeries of codimension 2 or higher, while keeping the Steklov spectrum nearly unchanged. This shows that certain changes in the topology of a domain do not have an effect when considering shape optimization questions for Steklov eigenvalues in dimension 3 and higher.

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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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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 - 2021)

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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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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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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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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