The eigenvalues of the Laplacian encode fundamental
geometric information about a Riemannian metric. As an
example of their importance, I will discuss how they
arose in work of Cao, Hamilton and Illmanan, together
with joint work with Stuart Hall,  concerning stability
of Einstein manifolds and Ricci solitons. I will outline
progress on these problems for Einstein metrics with
large symmetry groups. We calculate bounds on the first
non-zero eigenvalue for certain Hermitian-Einstein four
manifolds. Similar ideas allow us estimate to the
spectral gap (the distance between the first and second
non-zero eigenvalues) for any toric Kaehler-Einstein manifold M in
terms of the polytope associated to M. I will finish by
discussing a numerical proof of the instability of the
Chen-LeBrun-Weber metric.

We show that for an immersed two-sided minimal surface in R^3,
there is a lower bound on the index depending on the genus and number of
ends. Using this, we show the nonexistence of an embedded minimal surface
in R^3 of index 2, as conjectured by Choe. Moreover, we show that the
index of an immersed two-sided minimal surface with embedded ends is
bounded from above and below by a linear function of the total curvature
of the surface. (This is joint work with Otis Chodosh)

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We describe a short, direct, alternative to the DeTurck trick to prove the
uniqueness of solutions to a large class of curvature flows of all orders,
including the Ricci flow, the L^2 curvature flow, and other flows related
to the ambient obstruction tensor. Our approach is based on the analysis
of simple energy quantities defined in terms of the actual solutions to the
equations, and allows one to avoid the step -- itself potentially
nontrivial in the noncompact setting -- of solving an auxiliary parabolic
equation (e.g., a k-harmonic-map heat-type flow) in order to overcome the
gauge-invariance-based degeneracy of the original flow. We also
demonstrate that, by the consideration of a certain energy
quotient/frequency-type quantity, one can give a short and quantitative
proof (avoiding Carleman inequalities) of the global backward uniqueness of
solutions to a large class of these equations.

Motivated by the pluriclosed flow of Streets and Tian, we establish
Evans-Krylov type estimates for parabolic "twisted" Monge-Ampere
equations in both the real and complex setting. In particular, a bound
on the second derivatives on solutions to these equations yields bounds
on Holder norms of the second derivatives. These equations are
parabolic but neither not convex nor concave, so the celebrated proof of
Evans-Krylov does not apply. In the real case, the method exploits a
partial Legendre transform to form second derivative quantities which
are subsolutions. Despite the lack of a bona fide complex Legendre
transform, we show the result holds in the complex case as well, by
formally aping the calculation. This is joint work with Jeff Streets.