Mathematics Problem Archive

A distribution is singular transitive if any two points can be connected by a concatenation of singular curves. Does singular transitivity imply that ...

But deeper structures (if they exist at all) that lie at the roots of Dirac operators and of minimal hypersurfaces are yet to be revealed....

Identify the most general classes of geometric objects having properties analogous to those of $C^2$ Riemannian manifolds with $\operatorname{Sc}\geq\...

Question. What could be a, possibly non-geometric, extension of the concept ofSc ≥0, where one would be able perform symmetrization and reduce the cas...

Question. How common are Ricci flat metrics on compact simply connected manifolds X which admit metrics with positive scalar curvatures?...

Conjecture. Connected sums of sufficiently many copies of compact manifolds that are not homotopy spheres carry no Ricci-flat metrics....

Problem. Remove the spin an the uniform positivity conditions, relax completeness and determine the sharp value ofconstn (depending on theK-theory cla...

It seems not impossible, at least for compactX, that, in fact, K-area(X×R) =K-area(X×S1)....

Question. Is the residual finiteness of the fundamental group essential?...

(i) It is unclear if the last step in the above argument is truly needed: conceivably, maps Φ ∶Sn−1→U(N) with Lip(Φ) < 1 2 are contractible to constan...

(iii) TheSn- andS2-product inequalities seems to hold fornon-trivial sphere fibrations, but I have not checked this carefully....

On the other hand, if theδ-neighbourhoods Uδ(S) ⊂X of allT(X)- non-spin surfacesS in a Riemannian manifoldX are "large" then the spin area of X must b...

For instance, letX be homomorphic to CP 2 and letvol(Uδ(S)) ≥δ2 for allT(X)-non-spin surfacesS ⊂X and 0<δ ≤1. Is then spin-area(X) ≥1/1 000 000?...

On the other hand, there probably exist compact simply connected n-dimensional manifolds for alln ≥4 with arbitrarily prescribed (finite) values of the...

the sharp values ofwidthn−m for these solids remains problematic for m≥2, (unless I missed some paper)....

Are there compact manifoldsX which support metricsg withSc(g) > 0 but admit no area extremal or length extremal metricsg?...

Can one "effectively" evaluate the minimal constantλ=λ(X,g)„ such that a given Riemannin manifoldX = (X,g), e.g wheresect.curv(g) > 0, would support an...

But it is unclear if this remain true with "area" in place of "length"....

Question. When does such anX0 is area extremal in the category of complete manifolds?...

Extension Problem.LetX be a Riemanniann-manifold withSc(X) ≥σ > 0 and letσ−≤σ,r andr+ ≥r be positive numbers. Whendoesthereexistan n-dimensionalmanifo...

Conjecture. Completion by Extension.If σ > σ−and r ≥constn(σ −σ−)−1 2 for some (large) constant constn, then the extension problem is solvable withr+ ...

What are possible values of the co-s+areas of the complements ofr-balls in compact spin manifoldsX withSc(X) ≥n(n−1)?...

what is the (asymptotically forn→∞and/or fork→∞) sharp inequality for immersions of theseΣn−1 to spheres...

Is then every immersion fromXj to the unit ball in RN satisfies supcurv(Xj↪BN(1) ⊂RN)) ≥ √ k for allN ≥n1+....+nj+ 1?...

Question. Are all extremal convex polyhedraP are mean convexly extremal?...

Is the regular Euclidean $3$-simplex mean-convexly extremal? Equivalently, can a simplex mapped facewise to it without decreasing distances have nonne...

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?...

[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...

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

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

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....

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 $F_{12}$?...

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....

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...

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

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...