Mathematics Problem Archive

Probably, these equalities imply thatP is isometric to a Euclidean rectangular solidbut the approximation/smoothing is no good for proving this kind o...

This suggests a possibility of definingSc(X) ≥0 for some singular spaces, X, e.g. for manifolds with continuous (bounded measurable?...

Let $\widetilde X$ be the universal cover of a Riemannian $n$-manifold $X$ homeomorphic to the $n$-torus. Conjecture that $\widetilde X$ has non-posit...

Conjecture. Shrinking of Singularities. Let X be a compact orientable Riemanninn-manifold, f0 ∶X →Tn be a continuous map of non-zero degree, hi, i=0,1...

Conjecture Let a domainY ⊂Rn havemean.curv(∂Y ) ≥ n−k+ε for someε> 0 and k = 2,...,n −1. ThenY−1 admits a continuous map onto a(k−1)-dimensional polyh...

[a] Are the diameters diamc(Lip1(Bn(R)→S))) bounded for a large fixedc and R →∞if Hi(S, R) = 0 for i = 1, 2,...,n....

[b] What is the asymptotics of the diameters diamc(Lip1(Bn H(R)→S))) for the hyperbolic ballsBn H(R) and R→∞?...

[c] LetS be a Riemannian manifold homeomorphic to the connected sum of twenty copies ofS2× S2. Are there 1-Lipschitz maps fR ∶B4(R)→S, R→∞, such thath...

Conjecture. Parametric Hypersphericity. Let X be a complete oriented Riemanniann-manifold and letΨ(X) ⊂Lipλ(X →Sn(1))) be the space of 1-Lipschitz loc...

If m = n−1 then, conjecturally, this is the only manifold with this property: the inequalities macr.dim(Ψ(X)) ≥1 and Sc(X) ≥(n−1)(n−2) should imply th...

Conjecture. Stability of Periodic Slabs. The only Zn−3-invariantmeanconvexdomainsin Rn withdisconnectedboundaries are slabs between parallel hyperplan...

Problem. Describe "Remnants of Collapse" of Hypersurfaces with Scalar Curvatures Blowing-up to+∞. Namely, decide when a closed subsetY in aC2-smooth R...

Conjecture. Subsets with Low Hausdorff Dimensions are Remains of Scalar Curvature Blow-ups.All closed subset Y ⊂W with dimHau(Y ) <n−1=dim(W)−2, are in...

Conjecture. InvarianceandNon-invarianceof Sc∩(Y ) = +∞. The inequalitySc[n] g∩(Y ) =+∞is independent of the Riemannian metric g in W ⊃Y Moreover it is...

Conjecture. Stabilisation under Cartesian Products. [Sc[n] g∩(Y ) =+∞]⇔[Sc[n+k] g⊕gk∩(Y × Xk) =+∞], where Xk = (Xk,gk) is a compact Riemannian manifol...

Conjecture. C0-closeness of the spaces ofC0-metrics withVolumicallyPositiveScalarCurvatures. IfaRiemannian C0-metricg onan n-dimensionalmanifold X can...

Conjecture. C2-Smoothing of Continuous Metrics with Volumically Positive Scalar Curvatures.All continuous Riemannian metricsg on a smoothn-dimensional...

Conjecture. Topological Equivalence of Different Scalar Curvatures. If a smoothn-manifold admits acontinuous metricg1 withScvoln(g1) > 0 then it also a...

Conjecture. C0-ContinuousGuth-GerochLowerVolume Bound for Balls in the Coverings of Essential Manifolds. [57]. The universal coverings ˜X of Q-essenti...

Conjecture. Non-Riemannian Guth-Geroch. Let X be an n-dimensional Q-essential pseudomanifold (e.g. manifold) with an arbitrary metric. Then the univer...

Conjecture Non-Riemannianε-Llarull. Let a compact n-dimensionalpseudomanifoldhastheHilbertvolumesofallitsballs of radii≤ε0 smaller than the volumes of...

Conjecture. C0-Density of C0-metrics with Volumically Positive Scalar Curvatures. Continuous Riemannian metrics withScvoln > 0 on anX are dense in the...

Conjecture. Geroch for Alexandrov Spaces.If anndimensionalAlexandrovspace X withsect.curv ≥−1andScvoln(X) ≥ 0 admits a continuous mapΦ with non-zero d...

but there are no apparent examples (if any) where these inequalities are strict. Everything we know aboutK-area+ easily extends to the the FredholmKar...

Conjecture. Hyperbolic Volume Inequality.Then every continuous mapf0∶X→X0 is homotopic to a mapf, such that voln(f(X)) ≤vol(X) where, moreover, this i...

Prove that there is a dimension-dependent constant $c_n$ such that every compact Riemannian $n$-manifold $X$ with $\operatorname{Sc}(X)\geq-\sigma^2$ ...

Can one hear the presence of an orbifold singularity, i.e. whether or not there exists a pair of isospectral orbifolds, one of which has singular poin...

For every $\ell,k>0$, there exist $C,L,K>0$ with the following effect. Let $(M,g)$ be a complete Riemannian manifold with injectivity radius $inj(M,g)...

Give an alternative, geometric proof that the holonomy representation of a closed flat manifold is reducible....

Does $S^2\times T^2$ admit a Riemannian metric with positive biorthogonal curvature?...

1. Does ${\rm Sing}_b (T)$ have zero $(m-1)$-dimensional Hausdorff measure? 2. If yes, does ${\rm Sing}_b (T)$ have (Hausdorff) dimension at most $m-2...

Which manifolds can be modeled on an orbit of a real form in a space of flags?...

What is the homotopy classification of totally real immersions of real $3$-manifolds in the complex full flag manifold $F_{12}$?...

For ${2n+1}>3$ and $0<k<{2n+1}$, are projective subspaces obtained by projection of the $k$-dimensional equatorial spheres minimal submanifolds of the...

Assume that a left-invariant Riemannian metric is given on a compact connected Lie group $G$ such that the index of any geodesic segment is even. Must...

Let $M$ be a connected closed manifold with finite fundamental group of odd order. Assume that the universal cover of $M$ admits a metric of positive ...

Let $M$ be a connected closed manifold admitting a metric of positive scalar curvature. Does this imply that $M$ is $p$-atoral for all odd $p$?...

Find more numerical invariants of metric spaces that are nondecreasing under coarse embeddings....

Find applications of the harmonic map approximation of coarse embeddings....

For solvable Lie groups $G$, show that solvsolitons are the only local maxima of the Ricci pinching functional $g\mapsto F(g)=\frac{Scal(g)^2}{|Ric(g)...

Explain the existence and the role of the symplectic nature of the groupoid/algebroid and its relevance for the geometry of the moduli spaces of geome...

; Do there exist embedded minimal surfaces with finite genus and Morse index $4$?; More focussed: in the $1$-parameter deformation of Costa's surface,...

- Construct congruence lattices which are norm$_1$ and norm$_1*$- isospectral in all dimensions (see ). - Are there families of $p$-isospectral lens s...

Prove that geodesic spheres provide the least-perimeter way to enclose prescribed volume in $CP^2$....

Prove that the pictured standard triple soap bubble is the least-perimeter way to enclose and separate three given volumes in $\mathbb{R}^3$....

1. If the normal holonomy group of an irreducible and full homogeneous submanifold $M^n$ of the sphere with $n \geq 2$ does not act transitively, then...

; Do CMC spheres about a point $x$ in such a space form a foliation of $X-\{x\}$?; Could this be a way of proving embeddedness of CMC spheres in gener...

; Calabi-Yau problem. For an embedded minimal surface in $\mathbb{R}^3$, does complete imply proper?; Hoffman-Meeks conjecture. For a complete embedde...

Is every Laplacian algebra of polynomials maximal?...

Prove the lower bound \[ d\le \mbox{Index}(\Phi_{\mathcal A})+\mbox{Null}(\Phi_{\mathcal A}), \] where $\mbox{Null}(\Phi_{\mathcal A})$ is the {\it nu...