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It seems that people often talk of "doing Morse theory" on loop spaces in two quite different contexts. Case 1: When one does Morse theory on a loop space $\Omega(M; p,q)$ using the energy functiona …

Alexandrov's Theorem says that a compact constant mean curvature hypersurface embedded in $\mathbb{R}^{n+1}$ must be a round sphere. What happens when the mean curvature is small, or bounded? (For in …

Consider the round metric on $S^n$. The geodesics are (multiples of) great circles, and one can verify that this metric is of Morse-Bott type. The Morse indices of the n-covered great circles are (if …

The Gauss-Bonnet theorem characterizes the topology of surfaces by means of their Gaussian curvature. Do there exist results characterizing the topology of surfaces embedded in $\mathbf{R}^3$ via th …

Let $(M, g)$ be a compact Riemannian manifold. Let $i(-)$ be the index of a geodesic and let $l(-)$ be the length. Is there an inequality of the form $i(\gamma) \leq C l(\gamma)$ for some $C>0$ depend …

Fix an almost-complex structure $J$ on $\mathbb{R}^{2n}.$ Let $u: (D^2, i) \to (\mathbb{R}^{2n}, J)$ be a $J$-holomorphic disk. My question: can one prove an a-priori bound on the diameter of $u$ (s …