49 problems

Ricci-flat tangent cone conjecture. If one tangent cone at infinity is Ricci flat, then $M$ is Ricci flat. Consequently, the tangent cone at infinity should be unique; analogous qu…

Donaldson–Sun's metric-independence conjecture. Both $W$ and $\mathcal C$ depend only on the germ of the singularity $(Z,p)$, rather than on the metric.

Splitting conjecture. Every tangent cone at infinity $M_\infty$ splits isometrically as

Meeks's conjecture. Every embedded minimal surface in $\mathbb{R}^3$ with quadratic area growth has a unique tangent cone at infinity.

Let $Y$ be the object under consideration, let $p$ be a point, and let $TC_pY$ denote its tangent cone at $p$. Write $0_p$ for the zero element in this tangent cone, and let…

Let $Y$ be an exact special Lagrangian submanifold in $\mathbf C^{n+1}$ with tangent cone $C\times\mathbf R$ at $0$. Assume that $C$ is stable, meaning that the only linear and qua…

Let $(X^4,\mathsf{d})$ be a noncollapsed Ricci limit space, and consider the set of points whose tangent-cone cross-sections are not homeomorphic to $S^3$. Discrete singularity con…

Let $(X^n,\mathsf{d})$ be a noncollapsed Ricci limit space. For $k$-symmetry, require a tangent cone to split off an isometric factor $\mathbb{R}^k$ (as in the source). Naber's con…

Uniqueness conjecture. Tangent cones are unique for noncollapsed Einstein manifolds.

Consider the horospherical tangent cones at infinity of symmetric spaces, with restricted root systems whose factors determine the relevant types. Regularity conjecture. The horosp…

Let $(M,g,J)$ be a complete Calabi–Yau manifold with Euclidean volume growth, let $C$ be its unique asymptotic Calabi–Yau cone, and let $W$ be the central fibre of the associated…

Let $T^0_{\eta}(\mathcal{E})$ denote the candidate regular tangent cone to the admissible set $\mathcal{E}$ at $\eta$, and let … be the corresponding regular tangent cone in…

Let $(X_j,g_j,p_j)$ be a sequence of Einstein metrics converging with non-collapsing volume to a singular Einstein metric $(X_\infty,g_\infty,p_\infty)$, and let…

Let $(Z,\omega_Z,p_Z)$ be a bubble limit, let $\mathcal C_0(Z)$ be its unique tangent cone at $p_Z$, and let $W=\operatorname{Spec}(R(p_Z))$ be the Fano cone obtained from the asso…

Let $R={\mathbb C}[{\bf z}^{(v)}]$, let $I_{v,w}$ be the Kazhdan–Lusztig ideal defining the local chart of $X_w$ at $E^{(v)}_\bullet$, and let $\overline I_{v,w}$ be its ideal of l…

Let $X\subset\mathbb{C}^n$ be a pure $d$-dimensional entire complex analytic subset. Suppose that there exists a sequence $T=\{t_j\}_{j\in\mathbb{N}}$ of real positive numbers such…

Let a properly immersed minimal surface in $\boldsymbol{R}^3$ have quadratic area growth. Meeks' conjecture. Any such surface has a unique tangent cone at infinity. This is stated…

Unique-expansion conjecture. The elements of $\operatorname{Ap}(\Gamma_m,a)$ have unique expressions. Hence the tangent cone of $k[[\Gamma_m]]$ is Cohen–Macaulay. This would extend…

Let $\mathbf{C}^3$ carry a Calabi–Yau metric whose tangent cone at infinity is $\mathbf{C} \times A_2$. Identify metrics that differ by biholomorphism and scaling. Székelyhidi's pa…

All-symmetries conjecture. For each $k\leq n-2$, there exists $\hat{\mathcal{S}}^k$ with $\dim\hat{\mathcal{S}}^k\leq k$ such that for each $x\in X\setminus\hat{\mathcal{S}}^k$, ev…

Tangent-cone homeomorphism conjecture. The tangent cones of $X$ are all homeomorphic at a fixed point away from a set of codimension five.

Colding–Naber's tangent-cone uniqueness conjecture. The tangent cones of $X$ are unique away from a set of codimension three.

Tangent-cone uniqueness conjecture. There is a unique tangent cone to $T$ at $p$.

Let $A$ be an admissible Hermitian–Yang–Mills connection on $(\mathcal{E},B)$, and let $\hat{\mathcal{E}}$ be any chosen optimal extension of $\mathcal{E}$ at $0$. An analytic tang…

Let $B$ be a ball in $\mathbb C^n$ and let $\mathcal E$ be a reflexive sheaf over $B$ with an isolated singularity at $0$. For an admissible Hermitian–Yang–Mills connection $A$ on…