173 problems

A homogeneous expanding Ricci soliton is a homogeneous Riemannian manifold whose Ricci flow evolves by scaling and diffeomorphisms with an expanding scale factor. Generalized Aleks…

Let $M$ be a compact transverse Fano Sasakian manifold of dimension five, and let its Sasaki–Ricci flow be a solution of the flow equation referred to in the source. A shrinking Sa…

Let $(M,g(t))_{t\in[0,T)}$ be a solution to Ricci flow on a closed smooth $n$-dimensional Riemannian manifold, with $T$ its maximal time. Hamilton's scalar-curvature blow-up conjec…

The Ricci flow with surgery on a 3-dimensional manifold depends on a surgery scale $\delta>0$, which can be chosen arbitrarily small. Perelman's conjecture. As $\delta$ tends to ze…

Type I scalar-curvature conjecture. Finite-time singularities of Ricci flows on Kähler manifolds should be Type I at least in the sense of scalar curvature.

Perelman's ancient oval conjecture. Every compact ancient $\kappa$-solution must either be a quotient of a family of shrinking round spheres or be isometric to a so-called ancient…

A four-dimensional κ-noncollapsed steady gradient Ricci soliton is a complete steady gradient Ricci soliton satisfying the stated curvature conditions: it has nonnegative curvature…

Consider an ancient asymptotically cylindrical 4-dimensional Ricci flow, meaning a Ricci flow existing for all sufficiently negative times and asymptotic at negative infinity to a…

Hamilton's conjecture. At least in dimension four, every linearly stable $GRS^+$ metric is an Einstein metric with positive scalar curvature.

Let a manifold evolve under the Ricci flow, and consider the singular times at which the flow develops singularities. Hamilton's Ricci-flow surgery conjecture. The flow should brea…

Cao's conjecture. A Ricci shrinker has constant scalar curvature if and only if it is isometric to a finite quotient of

Combinatorial Ricci-flow conjecture. The following statements are equivalent: (1) $\mathbb{E}[L(\rho)]>0$; (2) $G$ is almost surely ICP-hyperbolic; and (3) there exists an initial…

Hamilton's conjecture. If $(M,g(0))$ is smooth, connected, complete and Ricci pinched, then it is either Riemannian flat or compact.

Initial-metric independence conjecture. The singularity type at infinity is independent of the initial metric $\omega_0$.

A uniformly Ricci pinched Riemannian manifold is a complete, connected 3-dimensional Riemannian manifold satisfying the uniform Ricci pinching condition. Hamilton–Lott conjecture.…

Short-time existence conjecture. There exists $T>0$ and a smooth family of complete metrics $g(t)$ on $M\times[0,T)$ such that $g(0)=g_0$ and $g(t)$ solves the Ricci flow equation…

Cheeger–Tian's epsilon-regularity conjecture. Such an $varepsilon$-regularity theorem should hold on higher-dimensional Einstein manifolds, $4$-dimensional shrinking solitons, Ricc…

Let $s$ be the trigonometric deformation parameter and let $tau$ denote RG time. A flow is called linear when $log(s)$ is proportional to $tau$. Ricci flow conjecture. For Hermitia…

Chen's Ricci-flow conjecture. The only obstruction to the long-time existence of Ricci flow is an $L^\infty$ bound on the scalar curvature.

Hamilton's Type-III asymptotic soliton conjecture. Type-III solutions asymptotically approach soliton metrics that are locally homogeneous.

In closed string theory, Ricci flow arises as a first-order world-sheet renormalization-group flow. Off-shell processes in string field theory, such as tachyon condensation, are co…

Let $(M,g(t))$ be a solution to the Ricci flow on a compact $3$-manifold with arbitrary initial metric. Hamilton's Harnack-estimate conjecture. A Harnack-type estimate holds for th…

Let a finite-time singularity of the Ricci flow on a compact $3$-manifold be dilated, and suppose a subsequential limit is a quotient by isometries of one of $S^3$,…

Let $(\mathbb S^{2},g(t))$ be a Ricci flow on the two-sphere, and let $\lambda_{1}(t)$ denote the smallest positive eigenvalue of the Laplacian of $g(t)$. Monotonicity conjecture.…

Let $R(x,t)$ denote the scalar curvature of the standard solution of the Ricci flow on $\mathbb{R}^3$, defined for $0\le t<1$. Perelman's scalar-curvature conjecture. There exists…