19 problems

Homeomorphism extension. The homeomorphism version of Theorem would also hold for any thin point $x$ with $f_*={\rm id}$ and for any $x\in X_0^2$ if the ball…

Let $F\colon X\to X'$ be a homotopy equivalence of manifolds with torsion-free fundamental groups that restricts to a diffeomorphism on the boundary. Equip the boundaries with Riem…

Let $M$ be a closed $(2k-1)$-manifold with torsion-free fundamental group, and let $\alpha$ be a unitary representation used to define the rho-invariant $\rho(M,\alpha)$. Weinberge…

Let $(M^n,g)$ be a complete Riemannian manifold with smooth nonempty boundary $\partial M$. Suppose $M$ has nonnegative Ricci curvature and $\partial M$ is uniformly convex, meanin…

Relative index comparison conjecture. There exists a map

Uniform a priori estimate conjecture. There exists $C(M,g,\delta)>0$ such that

Existence conjecture. Given $K<0$ and $c>\sqrt{-(n-2)K/n}$, there exists a smooth solution $u>0$ to equations.

Let $n\geq 2$ be such that the generalised Geroch conjecture without boundary holds for all $n$-manifolds. Let $M$ be an $n$-manifold with non-empty boundary, and form $M\# T^n$ by…

Boundary-strata conjecture. If $j$ is even, then $\beta|_{\mathsf{\int}(\partial_j^+X(v_\beta))}$ is a contact form away from a set of measure zero, and

Let $M^n$ be a compact $n$-manifold with non-empty boundary, and let its double be the closed manifold obtained by gluing two copies of $M^n$ along their common boundary. Let…

Let $X$ be a compact manifold with boundary, let $M=\operatorname{Dbl}(X,\partial X)$ be its double, and let a metric of positive scalar curvature mean a Riemannian metric whose sc…

Let $(M,g_0)$ be a smooth compact Riemannian manifold of dimension $n\geq 3$ with boundary $\partial M$. For constants $c_1,c_2\in\mathbb{R}$, consider the problem … where $u$ is r…

Let $(M,\partial M)$ be a compact aspherical $n$-manifold-with-boundary, and write $b^{(2)}_{>n/2}(M)=0$ to mean that all $L^2$-Betti numbers in degrees greater than $n/2$ vanish.…

Relative Gromov–Lawson–Rosenberg conjecture. If the relative higher index of $D$ is zero, then $N$ admits a metric of positive scalar curvature that is collared near $\partial N$.…

Generic-path conjecture. Given $v_0$ and $v_1$, there is a one-parameter family of traversing fields $\{v_t\}_{t\in[0,1]}$ connecting $v_0$ to $v_1$ such that…

Boundary-generic realization conjecture. The constraints

Disk obstruction conjecture. The standard $(n+1)$-disk $D^{n+1}$ does not admit a traversing boundary concave vector field.

Let $g$) be a smooth Riemannian metric defined on the unit half $n$-ball $B^+_1$, where $4\leq n\leq24$. Suppose there is a sequence of positive solutions $\{u_i\}$ of … with…

Let $M^m$ be a closed, smooth, oriented Riemannian manifold and let $N^n$ be a closed submanifold of codimension $m-n \geq 2$. Define the compact Riemannian manifold … where…