Let $(M, g)$ be a steady gradient Ricci soliton of dimension $n \geq 4$ which has positive sectional curvature and is asymptotically cylindrical. Under these assumptions, we show that $(M, g)$ is rotationally symmetric. In particular, our results apply to steady gradient Ricci solitons in dimension $4$ which are $\kappa$-noncollapsed and have positive isotropic curvature.

For several classes of second-order dispersionless PDEs, we show that the symbols of their formal linearizations define conformal structures that must be Einstein-Weyl in 3D (or self-dual in 4D) if and only if the PDE is integrable by the method of hydrodynamic reductions. This demonstrates that the integrability of these dispersionless PDEs can be seen from the geometry of their formal linearizations.

We establish an identity for closed hyperbolic surfaces whose terms depend on the dilogarithms of the lengths of simple closed geodesics in all 3-holed spheres and 1-holed tori in the surface.

The image of the Gauss map of any oriented isoparametric hypersurface in the standard unit sphere $S^{n+1}(1)$ is a minimal Lagrangian submanifold in the complex hyperquadric $Q_n(\mathrm{C})$. In this paper we show that the Gauss image of a compact oriented isoparametric hypersurface with $g$ distinct constant principal curvatures in $S^{n+1}(1)$ is a compact monotone and cyclic embedded Lagrangian submanifold with minimal Maslov number $2n / g$. We obtain the Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces of classical type with $g = 4$. Combining with our results in On Lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres and Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces II, we completely determine the Hamiltonian stability of the Gauss images of all homogeneous isoparametric hypersurfaces.

We show that for a manifold with non-negative curvature one obtains a collection of concave functions, special cases of which are the concavity of the length of a Jacobi field in dimension $2$, and the concavity of the volume in general. We use these functions to show that there are many cohomogeneity one manifolds which do not carry an analytic invariant metric with non-negative curvature. This implies in particular, that one of the candidates in Positively curved cohomogeneity one manifolds and 3-Sasakian geometry does not carry an invariant metric with positive curvature.