User jiangsaiyin - MathOverflow

simple explaination of simplicial volume=4g-4 when genus $\ge 1$

2013-05-22T09:32:56Z

In Gromov's famous book ,it says "simplical volume of every oriented surface of genus $ \ge 1$ satisfies${\left\| {\left[ S \right]} \right\|_\Delta } = 4g - 4 = - 2\chi \left( S \right) = - 2({k_0} - {k_1} + {k_2})$(Euler characteristic in this condition is negative).One proof is to use propertionality theorem for compact locally homogeneous space V,${\left\| {\left[ V \right]} \right\|_\Delta } = cvol\left( V \right)$,where the constant c depends on the local geometry of V.But Gromov says it's elementary!Since the simplicial volume is the infimum of the number of simplices over all homotopy triangulations of V.It's easy to see that "we need at least 4g-4 simplices to triangulate S?Or for a surface with negative Euler characteristic,${k_2} \ge - 2\chi \left( S \right)$?I can't see it.Please give an explaination.

What does a singular simplex with real coefficient mean

2013-05-21T16:13:51Z

For an $n$-dimensional orientable closed manifold $M$, the simplicial volume is the infimum of the $l^1$-norm of the elements $\sum a_i \sigma_i$ ($a_i \in \mathbb{R}$) which represent the fundamental class. My question is: since $\sigma_i$ is a continuous map from the $n$-dimensional simplex to $M$, what does $a_i \sigma_i$ mean? Is it the same kind of map? I don't know how to make sense of this expression.

fundamental class is the sum of simplices of triangulation of the manifold?

2013-05-21T14:46:33Z

M is an n-dimensional closed orientable manifold. I find in a book "Intuitively,the fundamental class can be thought of as the sum of the (top-dimension) simplices of a suitable triangulation of the manifold". My explanation is we can use the Mayer-vietoris sequence to "glue" the local orientations inductively over a finite oriented atlas together to construct the fundamental class. One open subset of the atlas correspond to one simplex. Am I right?

fundamental class and simplicial volume

2013-05-13T15:02:51Z

My topology is poor.In Gromov's famous book,it says"If h is the fundamental class of an orienatable manifold V,then the ${l^1}$ norm is obviously majorized by the number of the top-dimensional simplices needed to triangulate V,why?The simplicial volume of an oriented closed manifold is the ${l^1}$ norm of the fundametal class.The fundamental class of ${S^n}$(take it as the boundary of ${\Delta ^{n + 1}}$) is $ \pm \partial {\delta ^n}$,where ${\delta ^n}$ is the identity map on ${\Delta ^{n + 1}}$.So I think the simplicial volume of ${S^n}$ is the number of the n-dim simplex (ie boundary) of ${\Delta ^{n + 1}}$.When n=2,volume is 4.But I think it needs only 2 2-dim simplices to triangulate the 2-dim sphere.It's a surprise to me that the volume is 0.0 simplex can triangulate the sphere? Can someone tell me where I am wrong?

$\Delta f \le - \lambda f$ then ${\lambda _1}\left( M \right) \ge \lambda$?

2013-04-04T09:53:28Z

Let M be a complete Riemannian manifold.If there exists a positive function defined on M satisfying$\Delta f \le - \lambda f$ then ${\lambda _1}\left( M \right) \ge \lambda$?

curvature of cone over a cuboid bounded below?

2013-04-27T04:27:20Z

As we know,if M is an Alexandrov space with sec>=1,then the cone over M has sec>=0.What if when M is a cuboid with side length r1,...,rn,dia(M)<=π,then the cuvature of the cone over M bounded below by what?And the case when M is a ball in a hyperbolic space with dia(M)<=π?

examples of space of direction at a point in an infinite dim Alexandrov space compact

2013-04-22T09:33:41Z

The space of direction at a point in an infinite dim Alexandrov space can be compact?Please give examples or prove it's wrong.

homotopy from a lipschitz map on the boundary to a lip map

2013-04-22T04:08:49Z

f is a continuous map from a k-dim simplex E to a lipschitz contractible space U (the homotopy from the identity map to a fixed point map is lipschitz).Since U is lipschitz contractible,we can extend the restriction of f on the boundary of E to a lipschitz map g on the whole E.Then can we get a homotopy relative to the boundary of E from f to g?I think it's wrong,but on a paper I read it seems trivial.

Only finitely many fundamental groups in $M(n,k,v,D)$?

2013-04-18T08:08:51Z

Let $M(n,k,v,D)$ denote the class of compact manifolds with $Ric \ge \left( {n - 1} \right)k,vol \ge v,diam \le D$.In 1990,M.Anderson proved that "There are only finitely many fundamental groups among the manifolds in $M(n,k,v,D)$ for fixed n,k,v,D".(Peter Peterson "Riemannian geometry",page 277).The beginning of the proof is "Choose generators ${ {\gamma _1}, \ldots {\gamma _m}} ,d(x,{\gamma _i}(x)) \le 2diam(M)$ as in the lemma.Since the number of possible relations is bounded by ${2^{{m^3}}}$,we have reduced the problem to showing that m is bounded".Why is the number of possible relations bounded by ${2^{{m^3}}}$,and where does ${2^{{m^3}}}$ come from?The fundamental groups are finitely generated,and the numbers of generators have a uniform upper bound.Does this theorem means that the numbers of the elements of the fundamental groups have a uniform upper bound?(Although it's obviously wrong I think).

Is the first eigenvalue of a parabolic ends of a Riemanian manifold 0?

2013-03-14T09:28:00Z

Is the first eigenvalue of a parabolic ends of a Riemanian manifold 0?

noncompact manifold with two ends splits?

2013-04-05T13:19:33Z

Let M be a noncompact manifold with two ends.Then we can construct a line(geodesic for infinite time) on M.The Busemann function for the line $\beta $ is regular on M.By morse theory,M is homeomorphic to $R \times {\beta ^{ - 1}}\left( 0 \right)$.Then the metric on M can be written as ${g_M} = d{t^2} + {g_t} = d{t^2} + {\varphi ^2}\left( t \right){g_0}$.Where ${g_0}$ is the metric for ${\beta ^{ - 1}}\left( 0 \right)$.Can some one point it out where I am wrong and give a counterexample?

Is there a harmonic function on $M = R \times {}_{{e^{2t}}}N$

2013-04-04T13:11:58Z

M is a warped product $M = R \times {}_{{e^{2t}}}N$.N is the spherical suspension over$R{P^2}$.N is an Alexandrov space but not a manifold.So is M.Is there a harmonic function on M?What if we replace ${e^{2t}}$ by ${\cosh ^2}\left( t \right)$?

Level set of convex function on Riemannian manifold diffeomorphic to n-1 dim sphere?

2013-03-17T04:07:01Z

M is a Riemannian manifold,there exists a c>0,a 1-lipschitz function h:X→R,strictly convex on B(p,c) such that h(p)=0 is a strict local minimum of h.Then the level set {h=a} is diffeomorphic to the n-1 dim sphere?The subset {h<=a} diffeomorphic to n dim ball?

Integral of a harmonic function on a manifold with two non-parabolic ends

2013-03-14T09:19:36Z

Let M be a complete Riemannian manifold.Suppose there are two non-parabolic ends on M with respect to $M\backslash {B_p}\left( {{R_0}} \right)$Then there is a harmonic function f on M.Is it right that $\int_{{B_p}\left( R \right)} {f \le CR} $ for $R \ge {R_0}$ and a constant C independent of R?

Alexandrov space where a Yau's inequality that holds on Riemannian manifold fails

2013-03-14T08:24:27Z

Let M be a complete Riemannian manifold,suppose there is a harmonic function f on M.It was first observed by Yau that$${\left| {{\nabla ^2}f} \right|^2} \ge \frac{n}{{n - 1}}{\left| {\nabla \left| {\nabla f} \right|} \right|^2}$$The proof is as follow:By choosing an orthonormal basis ${e_1},{e_2}, \cdots ,{e_n}$ such that$\left| {\nabla f} \right|{e_1} = \nabla f,{e_\alpha }f = 0$for all $\alpha \ne 1$ we have$${\left| {{\nabla ^2}f} \right|^2} = \sum\limits_{i,j = 1}^n {f_{ij}^2} $$ $$ \ge \sum\limits_{j = 1}^n {f_{1j}^2} + \sum\limits_{\alpha = 2}^n {f_{\alpha 1}^2} + \sum\limits_{\alpha = 2}^n {f_{\alpha \alpha }^2} $$$$ \ge \sum\limits_{j = 1}^n {f_{1j}^2} + \sum\limits_{\alpha = 2}^n {f_{\alpha 1}^2} + \frac{1}{{n - 1}}{\left( {\sum\limits_{\alpha = 2}^n {{f_{\alpha \alpha }}} } \right)^2}$$$$ = \sum\limits_{j = 1}^n {f_{1j}^2} + \sum\limits_{\alpha = 2}^n {f_{\alpha 1}^2} + \frac{1}{{n - 1}}f_{11}^2$$$$ \ge \frac{n}{{n - 1}}\sum\limits_{j = 1}^n {f_{1j}^2} $$on the other hand,$$\begin{array}{l} 4{\left| {\nabla f} \right|^2}{\left| {\nabla \left| {\nabla f} \right|} \right|^2} = {\left| {\nabla \left( {{{\left| {\nabla f} \right|}^2}} \right)} \right|^2}\ = 4{\sum\limits_{j = 1}^n {\left( {\sum\limits_{j = 1}^n {{f_i}{f_{ij}}} } \right)} ^2}\ = 4f_1^2\sum\limits_{j = 1}^n {f_{1j}^2} \end{array}$$ hence$${\left| {{\nabla ^2}f} \right|^2} \ge \frac{n}{{n - 1}}{\left| {\nabla \left| {\nabla f} \right|} \right|^2}$$ I know this inequality does not hold on Alexandrov spaces,but I don't know how to construct a counterexample.Can someone find an Alexandrov space that this inequality fails for a positive measure points?And for a Riemannian manifold can we deform it to such an Alexandrov space?

Must a hyperbolic cone over Riemannian manifold be manifold?

2013-02-25T07:17:26Z

M is a hyperbolic cone over an n-1 dim Riemannian manifold N with $Ric(N) \ge - \left( {n - 2} \right)$ ie $M = R \times {}_{\cosh \left( t \right)}N$,Surely N is an Alexandrov space,must M be a manifold?must N be totally geodesic in M?If not,please give counterexamples and point out the singular points.

How to compute the first eigenvalue of hyperbolic space ${H^2}$and ${H^n}$?

2013-02-19T07:41:31Z

How to compute the first eigenvalue of hyperbolic space ${H^2}$and ${H^n}$?

Why is the Bochner formula on an Alexandrov space worse than on a Riemannian manifold?

2013-01-22T18:56:08Z

M is a Riemannian manifold with $$Ric \ge - (n - 1)$$,f is a harmonic function on M,let $$h = |\nabla f|$$.By choosing an orthonormal basis such that $$|\nabla f|{e_1} = \nabla f,{e_\alpha }f = 0,\alpha \ne 1$$,we can get $$|{\nabla ^2}f{|^2} \ge {\frac{{n|\nabla |\nabla f||}}{{n - 1}}^2}$$.And by bochner formula$$\Delta {h^2} = 2Ri{c_M}(\nabla f,\nabla f) + 2|{\nabla ^2}f{|^2}$$,we finally get$$\Delta {h^2} \ge - 2(n - 1){h^2} + \frac{{2n}}{{(n - 1)}}|\nabla h{|^2}$$ If M is an Alexandrov space with$$ \sec \ge - 1$$,Given a function$$u \in W_{loc}^{1,2}\left( \Omega \right)$$we define a functional Δ on Lip0(Ω) by$${L_u} \left( \phi \right) = - \int_\Omega {\left\langle {\nabla u,\nabla \phi } \right\rangle } dvol,\forall \phi \in Li{p_0}\left( \Omega \right)$$ Suppose$$f \in W_{loc}^{1,2},{L_f}= 0$$$$|\nabla f| \ge c > 0$$for some constant c,we can get$$ \Delta {h^2} \ge - 2(n - 1){h^2} + \frac{{2n}}{{(n - 1)}}{\left( {\frac{{\left\langle {\nabla f,\nabla h} \right\rangle }}{h}} \right)^2}$$due to the paper"sharp spectral gap and Li-Yau's estimate on Alexandrov space",which is worse than in Riemannian case because$$ \frac{{\left\langle {\nabla f,\nabla h} \right\rangle }}{h} \ne |\nabla h|$$ So why does this happen?Can you give an Alexandrov space where The Bochner formula for Riemannian manifold is not valid?

How to compute the first eigenvalue of \[M = R{ \times _{\cosh t}}{S^2}\left( {\frac{1}{2}} \right)\]

2013-01-28T13:41:23Z

$$M = R \times N$$with the warped product metric$$d{s^2} = d{t^2} + {\cosh ^2}\left( t \right)ds_N^2$$$$N = {S^{2}}\left( {\frac{1}{2}} \right)$$2-dim sphere with radius 1/2

How to compute the first eigenvalue of \[M = R \times {}_{\cosh t}N\]

2013-01-25T13:40:09Z

$$M = R \times N$$with the warped product metric$$d{s^2} = d{t^2} + {\cosh ^2}\left( t \right)ds_N^2$$where N(dimN=n-1) is a compact manifold with $$Ric \ge - \left( {n - 2} \right)$$It should be mentioned that M may not be a Riemannian manifold but an Alexandrov space.So how to compute the first eigenvalue of M?If we restrict to the case $$N = {S^{n - 1}}\left( {\frac{1}{2}} \right)$$an n-1 dim sphere with radius 1/2,then the result?

classification of non-compact Riemannian manifold with Ric>=-(n-1),and first eigenvalue λ=(n-1)^2/4

2013-01-21T05:59:12Z

If M is a non-compact Riemannian manifold with Ric>=-(n-1),we know that the first eigenvalue λ(M)<=(n-1)^2/4.What if λ(M)=(n-1)^2/4,then M would be?Are there any paper on this topic?

level set of function f coincides with that of |▽f|,M splits?

2013-01-17T07:42:57Z

M is a complete Riemannian manifold,f is a function on M with no critical points.If the level set of f coincides with the level set of |▽f|,then M must be topologically the product RXN?

examples of Alexandrov space with sec>=-1 and first eigenvalue =(n-1)^2/4

2013-01-15T16:46:47Z

could someone give some examples :nonRiemannian manifold Alexandrov space with sec>=-1 and the first eigenvalue equal to (n-1)^2/4

about parabolic cone

2012-10-23T06:01:16Z

I want to prove some Alexandrov space M is parabolic cone X x R.Since Alex has no Riemannian metric,so how to do?Is there any (triangle) formula about the relation of distance of two points in M and distance of the projection points in X?

how many nonparabolic ends guarantee a nonconstant harmonic function on Riemannian manifold?

2012-12-22T09:14:59Z

M is a Riemannian manifold.An end E is said to be a non-parabolic end if it admits a positive Green's function with Neumann boundary conditon on E.Otherwise,it is a parabolic end.If M has more than two nonpara ends,then there is a nonconstant harmonic function on M. What if M has one nonpara end,one para end,is there a nonconstant harmonic function on M?

Does this PDE has a general solution?

2012-11-18T04:31:43Z

$$K\frac{\partial }{{\partial x}}(h\frac{{\partial h}}{{\partial x}}) = \mu \frac{{\partial h}}{{\partial t}}$$ K and u are constants. If no,how to get a asymptotic solution?ie,linearize

distance formula for warped product

2012-11-01T05:23:07Z

M is warped product RXN.gM and gN are metric of M and N.gM=dt^2+exp(-2t)gN.What is the relation between distance of any two points in M and distance of their projection in N?ie,the distance formula.

Is level set of Busemann function on Alexandrov space again Alexandrov space?

2012-10-22T08:03:16Z

M is an Alexandrov space with curv>=-1,containing a line(ray).Is level set of Busemann function on M again Alexandrov space?If not,can you give a counterexample?

laplacian of Busemann function on hyperbolic space

2012-10-15T08:56:40Z

M is a hyperbolic space $\mathbb{H}^n.$ $secM=-1.$ $γ(t) : R \rightarrow M$ a line.Let $b_+$ be the Busemann function for $γ : [0,\infty) \rightarrow M,$ and $b_−$ the Busemann function for $γ : (−\infty, 0] \rightarrow M.$

Thus, $b_+(x) = \lim_{t\rightarrow \infty}(d(x, γ(t)) − t),b_−(x) = \lim_{t\rightarrow \infty}(d(x, γ(−t)) − t).$ I want to compute $\Delta b_+,\Delta d_-.$

Below is my computation,I don't know whether it's right. Δb+<=limt→+∞(n-1)cotht=n-1 Δb-<=limt→-∞(n-1)cotht=-(n-1) so Δ(b++b−)<=0,and (b++b−)(γ(t))=0,(b++b−)(M)>=0, so can I claim (b++b−)(M)=0? (Ric>=0,we can use mean value property for supharmonic function to prove the claim,but now Ric=-(n-1)) then get Δb+=n-1,Δb-=-(n-1).

Busemann function on Hyperbolic space

2012-10-15T13:28:26Z

Possible Duplicate:
laplacian of Busemann function on hyperbolic space

What's the laplacian of the Buseman function on Hyperbolic space H^n?=n-1?When restricted to geodesics,is it linear?And the level sets are totally geodesic?

Comment by jiangsaiyin

2013-05-22T10:09:42Z

@Lee:Just consider the case when the communicative ring with unit is the real line R.The module is a map from R$ \times $set of singular simplices to set of singular simplices.If${a_i}{\sigma _i}$is not singular simplex,how can it be a R-module?Since the fundamental class$\alpha $is the generator of${H_n}(M;R) \cong R$.But for R,the generator can be any element except 0.So$c\alpha $ is the fundamental class when c$ \ne 0$?,thus the simplicial volume is 0 when we let c approach 0?I know it's wrong,but I don't know where

Comment by jiangsaiyin

2013-05-22T10:03:19Z

Just consider the case when the communicative ring with unit is the real line R.The module is a map from R$ \times $ set of singular simplices to set of singular simplices.If ${a_i}{\sigma _i}$ is not singular simplex,how can it be a R-module?Since the fundamental class$\alpha $ is the generator of${H_n}(M;R) \cong R$.But for R,the generator can be any element except 0.So $c\alpha $ is the fundamental class when c$ \ne 0$?,thus the simplicial volume is 0 when we let c approach 0?I know it's wrong,but I don't know where

Comment by jiangsaiyin

2013-04-22T06:47:43Z

Can you explain it in detail?The difficulty is the homotopy relative to the boundary of E

Comment by jiangsaiyin

2013-04-18T12:47:24Z

I mean we only consider the generator you mentioned,for any γi,γj,there exists a γk such that γiγj=γk?

Comment by jiangsaiyin

2013-04-18T11:21:36Z

"all the relations are of the form γiγj=γk"means for any γi,γj,there exists a γk such that γiγj=γk?

Comment by jiangsaiyin

2013-04-11T14:21:00Z

The way you extend is not a deformation retract I think.We should fix ${B^2} \times 0 \cup {S^1} \times I$.Project from a point just outside the cylinder will do.

Comment by jiangsaiyin

2013-04-11T14:19:40Z

Oh,I know ,thank you.

Comment by jiangsaiyin

2013-04-11T09:43:54Z

But where is the middle point of Ex1 projected?

Comment by jiangsaiyin

2013-03-14T12:01:29Z

If one parabolic end,one non-parabolic end,the inequality holds?

Comment by jiangsaiyin

2013-02-26T03:17:12Z

It must be a manifold,please help me to close the question if you see it.

Comment by jiangsaiyin

2013-02-25T09:28:59Z

Sorry,I should write$M = R \times {}_{\cosh \left( t \right)}N$

Comment by jiangsaiyin

2013-01-21T06:57:19Z

Thank you!I should add "M is non-compact"

Comment by jiangsaiyin

2013-01-16T02:39:01Z

"sharp spectral gap and Li-Yau's estimate on Alexandrov spaces" ,the first page you can see the definition

Comment by jiangsaiyin

2012-12-31T02:07:22Z

For a Riemannian manifold M with Ric≥-(n-1),if there is a nonconstant,bounded harmonic function f with finite Dirichlet integral on M.Then M=RXN with the warped product metric ds^2(M)=dt^2+cosh^2(t)ds^2(N) N is a compact manifold with Ric≥-(n-2) due to Peter Li and jiawang Ping in"complete manifolds with positive spectrum.The main obstacle in Alexandrov space is that the bochner formula is not so good as in Riemannian manifold.

Comment by jiangsaiyin

2012-12-21T15:49:05Z

I mean why not let the integration equal to 1?