Let $M$ be an irreducible Riemannian symmetric space. The index $i(M)$ of $M$ is the minimal codimension of a totally geodesic submanifold of $M$. In [1] we proved that $i(M)$ is bounded from below by the rank $\mathrm{rk}(M)$ of $M$, that is, $\mathrm{rk}(M) \leq i(M)$. In this paper we classify all irreducible Riemannian symmetric spaces $M$ for which the equality holds, that is, $\mathrm{rk}(M) = i(M)$. In this context we also obtain an explicit classification of all nonsemisimple maximal totally geodesic submanifolds in irreducible Riemannian symmetric spaces of noncompact type and show that they are closely related to irreducible symmetric $\mathrm{R}$-spaces. We also determine the index of some symmetric spaces and classify the irreducible Riemannian symmetric spaces of noncompact type with $i(M) \in \{4, 5, 6 \}$.

We give a complete proof of the fact that a contact structure that is sufficiently close to a Reebless foliation is universally tight.

We show that for every positive integer n, there exists an explicit positive integer cn that only depends on n such that if $M$ and $M^{\prime}$ are canonically polarized complex projective manifolds of dimension $n$ and if $H^0 (M, mK_M)$ and $H^0(M^{\prime}, mK_{M^{\prime}})$ are linearly isometric with respect to the pseudonorm $\langle\!\langle\ \rangle\!\rangle$ for some $m \geqslant c_n$, then $M$ and $M^{\prime}$ are isomorphic. This generalizes a result of Royden for compact Riemann surfaces of genus greater than or equal to $2$. The same approach is used to prove similar and weaker results for projective manifolds with nonnegative Kodaira dimension. We also introduce a kind of singular index for singularities of pairs that refines the traditional log canonical threshold.

We prove the existence of at least two distinct closed geodesics on a compact simply connected manifold $M$ with a bumpy and irreversible Finsler metric, when $H^* (M; \mathbf{Q}) \cong T_{d,h+1} (x)$ for some integer $h \geq 2$ and even integer $d \geq 2$. Consequently, together with earlier results on $S^n$, it implies the existence of at least two distinct closed geodesics on every compact simply connected manifold $M$ with a bumpy irreversible Finsler metric.

We study the short-time existence and regularity of solutions to a boundary value problem for the Ricci–DeTurck equation on a manifold with boundary. Using this, we prove the short-time existence and uniqueness of the Ricci flow prescribing the mean curvature and conformal class of the boundary, with arbitrary initial data. Finally, we establish that under suitable control of the boundary data the flow exists as long as the ambient curvature and the second fundamental form of the boundary remain bounded.

A tautological system, introduced in [20][21], arises as a regular holonomic system of partial differential equations that governs the period integrals of a family of complete intersections in a complex manifold $X$, equipped with a suitable Lie group action. A geometric formula for the holonomic rank of such a system was conjectured in [5], and was verified for the case of projective homogeneous space under an assumption. In this paper, we prove this conjecture in full generality. By means of the Riemann–Hilbert correspondence and Fourier transforms, we also generalize the rank formula to an arbitrary projective manifold with a group action.