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 ...

[∗] no closed aspherical15 manifold admits a metric withSc > 0....

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...

Problem. Evaluate σ○(X0) and σ◻(X0) for "simple" Riemannian manifolds X0 = (X,g 0). ###◻Dirac operators, because they are invariant under isometries, ...

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...

Conjecture. Bounds on Width and on the Macroscopic Dimension. Complete n-dimensional Riemannian manifoldsX with the scalar curvaturesSc(X) ≥σ > 0 sati...

Conjecture. Bound on the Filling Radius forSc ≥σ > 0., fil.rad[X]≤constn⋅( inf x∈X Sc(X)(x))−2. This, in view ofA,B,C from the previous section, yield...

Conjecture. Asphericity⇒K-Area =∞. The universal coverings ˜X of compact aspherical manifoldsX satisfy K-area( ˜X) =∞. Notice that this inequality, ev...

Let $B=B\Gamma$ be the classifying space of a discrete countable group, and let $f:X\to B$ be a continuous map from a Riemannian manifold. Does there ...

Conjecture Area Extremality and Rigidity of Symmetric and Einstein Spaces. All Riemannin manifolds with positive and parallel Ricci tensor, in particu...

For instance, ifX =SO(n) withn≥5, then no known method can rule out metricsg ≥g on X with Sc(g) > Sc(g)....

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?...

Spin Problem. All of the above only applies to spin maps f ∶X→X, for which the required twisted Dirac operator defined, and, as on similar occasions we...

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,...

Conjecture. All of the above is satisfied, modulo constants, for alln-manifolds, possibly incomplete and/or with boundaries, withSc(X) ≥σ > 0. Namely m...

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. Sharp Spherical Length Comparison Inequality. Spheres with finitely many punctures are length extremal. In fact – this is, probably equival...

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

Conjecture 18. Interior Hemi-Spherical Area Inequality. The r-interiors of all compact Riemanninn-manifolds X with boundaries and withSc(X) ≥Sc(Sn) =n...

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

there is no apparent non-trivial bound on the width ofX = Σn−1× [−1, 1]even we assume that thesectional curvatureof X is = 1....

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

Also it is unclear if there are (non-trivial) inequalities of this kind for other exotic spheres....

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

But it is also not impossible that all manifolds admit immersions into the unit ball in the Hilbert spaceR∞with principal curvatures bounded by a univ...

Problem. Identify combinatorial typesPextr of convex polyhedra where all representativeP ∈P are extremal and also describe extremal P of non-extremal ...

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...