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Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
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Special Riemannian metrics on $n$-torus
Let $(T^n, g_0)$ be the flat $n$-torus. For every $c>0$, it is known that a small $C^{\infty}$-perturbation of $g_0$ produces a metric $g$ such that all closed geodesics of length less or equal to $c$ …
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Diameter of pseudoholomorphic curves
You can Riemann mapping theorem to create a holomorphic disk of any diameter in case $n=2$.