Mathematics Problem Archive

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)....

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 ...

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...

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+ ...

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...

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?...

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...

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

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

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....