Mathematics Problem Archive

Conjecture: Q-Non-Essentiality of Manifolds with Sc > 0. No rational homology class14 in the classifying spaceBΓ of a discrete groupΓ can be realised ...

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....

Conjecture. Singularities are Unstable. Brian White told me about 30 years ago that he believed that Volume minimising hypersurfaces in generic Rieman...

Conjecture 6. ISC: Singularities are Irrelevant. Schoen and Yau announced 35 years ago [110], [114] that their descent metod extends to singular minim...

Let $X_{\mathrm{fl}}=\mathbb{R}^n/\Gamma$ be a complete flat manifold whose group $\Gamma$ acts by parallel translations. If a complete Riemannian man...

Probably, one can fully determine assumptions onπ1(X) depending on Xfl needed for this conclusion for all flat manifoldsXfl....

Also one can possibly relax theisometry at infinitycondition by some "asymptotic flatness" and negativity of a suitable "energy at infinity"....

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?...

Besides, similar results are expected for allsingular Alexandrov spaces with lower curvature boundsbut the Almgren’s regularity theory has not been de...

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)....

Conjecture: Waist-Width Inequality. All complete Riemannian n-manifolds X satisfy widthn−1(X) ≤constn⋅waistn−k+1(X). Contractibility Radius. This "rad...

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...

Would it be more prudent to replace the conditionSc(g) > 0 byRicci> 0?...

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

Conjecture: Stabilisation of Extremality.Let X0 be a compact area extremal Riemannin manifold. Then A. X0× Rm is area gap extremal for allm. 53 B. X0×...

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

Conjecture. The sphereSn minus Σo is area extremal in the "subcomplete" sense for all closed subsetsΣo ⊂Sn of topological dimensions k ≤1. 20 Lengths,...

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+ ...

Conjecture: ExtremalityofConcaveSphericalBalls. The balls B(R) ⊂Sn of radiiR≥π 2 are length extremal: no Riemannian metricg on such a ball which is gr...

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

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?...