# Ailana Fraser

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## Graduate Student Supervision

##### Doctoral Student Supervision

Dissertations completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest dissertations.

We consider free boundary minimal submanifolds in geodesic balls of simply connected space forms and relate them to the eigenvalues and eigenfunctions of the Dirichlet-to-Neumann map for the Helmholtz equation, generalizing a well-known connection from Euclidean space. This relation is applied to derive bounds on an isoperimetric ratio on such submanifolds as well as to study their balancing. We begin work towards an extremal eigenvalue problem by showing that some normalization of the eigenvalues is needed before considering a maximization problem analogous to that of Laplace eigenvalues on closed surfaces and of Steklov eigenvalues on compact surfaces with boundary. The first variation of the second eigenvalue of the Dirichlet-to-Neumann map for the Helmholtz equation along a family of metrics is calculated, extending earlier calculations of Berger \cite{berger_1973} and Fraser and Schoen \cite{fraser_schoen_2016}. We also compute the spectrum of the Dirichlet-to-Neumann map for the Helmholtz equation on Euclidean annuli. Finally, we study free boundary minimal surfaces of revolution in the sphere $\mathbb{S}^3$ and hyperbolic space $\mathbb{H}^3$. By looking at two distinct one-parameter families of complete minimal surfaces of revolution in $\mathbb{H}^3$, we prove that one contains a one-parameter family of free boundary minimal annuli and the other contains only the free boundary totally geodesic disks. In $\mathbb{S}^3$, we find that a one-parameter family of complete minimal surfaces of revolution contains many free boundary minimal annuli, which may be embedded or have self-intersections, contained in a geodesic ball of $\mathbb{S}^3$ or not. We conclude by showing that at least some of the free boundary minimal annuli we found are embedded and contained in a geodesic ball which may be smaller than, equal to or greater than a hemisphere.

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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

Theses completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest theses.

The purpose of this thesis is to explore the properties of closed curves which are length-minimizing in a nontrivial homotopy class of a compact Riemannian manifold with boundary. This project was inspired by a question asked by Professor Liam Watson about the existence of length-minimizing curves on a torus with finitely many disks removed, and whether these curves leave the boundary components tangentially. In our research, we extend the question to compact smooth Riemannian manifolds with smooth boundary. We first discuss some preliminary results to determine an appropriate class of curves to minimize over. We then explore the properties of the length and energy functionals, showing that a minimizer of the energy is also a minimizer of the length. We directly minimize the energy functional to show the existence of a length-minimizing curve in any nontrivial homotopy class of a compact Riemannian manifold with boundary. Finally, we address the regularity of length-minimizing curves, showing that they are piecewise geodesics (possibly with infinitely many pieces).

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In this thesis, we explore a famous theorem of Schoen and Yau stating that there exists no metric of positive scalar curvature on the n-torus Tⁿ, for n ≤ 7. The proof is by induction: One first assumes a metric of positive scalar curvature on Tⁿ. Then, by applying techniques from geometric measure theory, the direct method of the calculus of variations yields an area minimizing hypersurface in each non-trivial homology class χ ∈ Hn-1(Tⁿ; ℤ). Using a stability argument, it is then shown that the induced metric on this hypersurface is conformal to a metric of positive scalar curvature, contradicting the inductive assumption.

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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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