33 problems

Let $M$ be a compact oriented smooth manifold with boundary. Its relative higher signatures are the corresponding signature invariants for manifolds with boundary. Relative Novikov…

Positive mass theorem. If $R_g,H_g\geq 0$, then $m(g)\geq 0$, and equality holds if and only if $M$ is isometric to $\mathbb{R}^n_+$. This is the expected positive mass statement n…

Schroeder–Strake conjecture. Under these assumptions, $(M^n,g)$ is flat.

Extension conjecture. The formula of Theorem 2.5 should also hold for any even-dimensional compact spin manifold with boundary.

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

Let $M$ be a compact manifold with boundary, and let $DM$ denote its double, obtained by gluing two copies of $M$ along their common boundary. A Riemannian metric on $M$ has positi…

Relative index comparison conjecture. There exists a map

Let $A$ be a compact Riemannian manifold-with-boundary of dimension $d\geq 2$ and Riemannian volume $v(A)\in(0,\infty)$. The paper's coverage quantities and theorems are originally…

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…

Consider Conner–Floyd's notion of bordism for manifolds with boundary, in which the Möbius strip is nonbounding. Conner–Floyd bordism conjecture. This is the correct kind of bordis…

Let $M$ be a connected smooth manifold with boundary of dimension $m\neq 3$. For each subset $\mathcal{X}\subseteq\pi_{0}(\partial M)$, let $\mathfrak{t}_{co}^{\mathcal{X}}$ be the…

Let $(M,g)$ be a compact Riemannian manifold with boundary, let $Y(M,\partial M,[g])$ denote the Yamabe invariant, and consider Equation … Y(M,partial M,[g])0, … admits a positive…

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 $M$ be a compact oriented manifold with non-empty boundary, and let $H^*(M)$ denote its cohomology, regarded as an algebraic structure of the type induced by the de Rham comple…

Smoothness conjecture for holography. The hypotheses in the second bullet of the Holography Theorem are superfluous: the conjugating homeomorphism $\Phi:X_1\to X_2$ is always a dif…

Let $M$ be a compact Riemannian manifold with boundary $\partial M$, let $q_0\in\partial M$ be a blow-up point, let $\gamma$ be the linear perturbation term, and let the trace-free…

For $n\geq 3$, let $\mathbb{R}^n_+=\\{y\in\mathbb{R}^n:y^n>0\\}$. Let $(N,g)$ be an asymptotically flat Riemannian manifold with non-compact boundary and decay order $p>(n-2)/2$, m…

Let $M_1$ and $M_2$ be manifolds whose states $psi_{M_1}$ and $psi_{M_2}$ are glued along a common boundary submanifold $\Sigma$ to form a state $\psi_M$. Let $\Omega_1$ and…

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 $\partial N$.…