User jiangsaiyin - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T11:57:11Z http://mathoverflow.net/feeds/user/24637 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/131428/simple-explaination-of-simplicial-volume4g-4-when-genus-ge-1 simple explaination of simplicial volume=4g-4 when genus $\ge 1$ jiangsaiyin 2013-05-22T09:32:56Z 2013-05-22T10:21:59Z <p>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. </p> http://mathoverflow.net/questions/131356/what-does-a-singular-simplex-with-real-coefficient-mean What does a singular simplex with real coefficient mean jiangsaiyin 2013-05-21T16:13:51Z 2013-05-21T19:07:43Z <p>For an $n$-dimensional orientable closed manifold $M$, the <em>simplicial volume</em> 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.</p> http://mathoverflow.net/questions/131347/fundamental-class-is-the-sum-of-simplices-of-triangulation-of-the-manifold fundamental class is the sum of simplices of triangulation of the manifold? jiangsaiyin 2013-05-21T14:46:33Z 2013-05-21T16:19:05Z <p>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? </p> http://mathoverflow.net/questions/130493/fundamental-class-and-simplicial-volume fundamental class and simplicial volume jiangsaiyin 2013-05-13T15:02:51Z 2013-05-13T15:02:51Z <p>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?</p> http://mathoverflow.net/questions/126495/delta-f-le-lambda-f-then-lambda-1-left-m-right-ge-lambda $\Delta f \le - \lambda f$ then ${\lambda _1}\left( M \right) \ge \lambda$? jiangsaiyin 2013-04-04T09:53:28Z 2013-05-07T18:22:00Z <p>Let M be a complete Riemannian manifold.If there exists a positive function defined on M satisfying$\Delta f \le - \lambda f$ then ${\lambda _1}\left( M \right) \ge \lambda$?</p> http://mathoverflow.net/questions/128887/curvature-of-cone-over-a-cuboid-bounded-below curvature of cone over a cuboid bounded below? jiangsaiyin 2013-04-27T04:27:20Z 2013-04-27T04:27:20Z <p>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)&lt;=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)&lt;=π?</p> http://mathoverflow.net/questions/128329/examples-of-space-of-direction-at-a-point-in-an-infinite-dim-alexandrov-space-com examples of space of direction at a point in an infinite dim Alexandrov space compact jiangsaiyin 2013-04-22T09:33:41Z 2013-04-22T17:13:18Z <p>The space of direction at a point in an infinite dim Alexandrov space can be compact?Please give examples or prove it's wrong.</p> http://mathoverflow.net/questions/128307/homotopy-from-a-lipschitz-map-on-the-boundary-to-a-lip-map homotopy from a lipschitz map on the boundary to a lip map jiangsaiyin 2013-04-22T04:08:49Z 2013-04-22T04:08:49Z <p>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.</p> http://mathoverflow.net/questions/127934/only-finitely-many-fundamental-groups-in-mn-k-v-d Only finitely many fundamental groups in $M(n,k,v,D)$? jiangsaiyin 2013-04-18T08:08:51Z 2013-04-18T18:42:11Z <p>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).</p> http://mathoverflow.net/questions/124489/is-the-first-eigenvalue-of-a-parabolic-ends-of-a-riemanian-manifold-0 Is the first eigenvalue of a parabolic ends of a Riemanian manifold 0? jiangsaiyin 2013-03-14T09:28:00Z 2013-04-11T15:22:00Z <p>Is the first eigenvalue of a parabolic ends of a Riemanian manifold 0? </p> http://mathoverflow.net/questions/126615/noncompact-manifold-with-two-ends-splits noncompact manifold with two ends splits? jiangsaiyin 2013-04-05T13:19:33Z 2013-04-05T14:57:48Z <p>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?</p> http://mathoverflow.net/questions/126510/is-there-a-harmonic-function-on-m-r-times-e2tn Is there a harmonic function on $M = R \times {}_{{e^{2t}}}N$ jiangsaiyin 2013-04-04T13:11:58Z 2013-04-04T13:20:24Z <p>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 ${\cosh ^2}\left( t \right)$?</p> http://mathoverflow.net/questions/124746/level-set-of-convex-function-on-riemannian-manifold-diffeomorphic-to-n-1-dim-sphe Level set of convex function on Riemannian manifold diffeomorphic to n-1 dim sphere? jiangsaiyin 2013-03-17T04:07:01Z 2013-03-17T04:07:01Z <p>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&lt;=a} diffeomorphic to n dim ball?</p> http://mathoverflow.net/questions/124487/integral-of-a-harmonic-function-on-a-manifold-with-two-non-parabolic-ends Integral of a harmonic function on a manifold with two non-parabolic ends jiangsaiyin 2013-03-14T09:19:36Z 2013-03-14T10:25:10Z <p>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? </p> http://mathoverflow.net/questions/124485/alexandrov-space-where-a-yaus-inequality-that-holds-on-riemannian-manifold-fails Alexandrov space where a Yau's inequality that holds on Riemannian manifold fails jiangsaiyin 2013-03-14T08:24:27Z 2013-03-14T08:24:27Z <p>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?</p> http://mathoverflow.net/questions/122861/must-a-hyperbolic-cone-over-riemannian-manifold-be-manifold Must a hyperbolic cone over Riemannian manifold be manifold? jiangsaiyin 2013-02-25T07:17:26Z 2013-02-25T09:28:00Z <p>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.</p> http://mathoverflow.net/questions/122276/how-to-compute-the-first-eigenvalue-of-hyperbolic-space-h2and-hn How to compute the first eigenvalue of hyperbolic space ${H^2}$and ${H^n}$? jiangsaiyin 2013-02-19T07:41:31Z 2013-02-19T07:54:39Z <p>How to compute the first eigenvalue of hyperbolic space ${H^2}$and ${H^n}$?</p> http://mathoverflow.net/questions/119592/why-is-the-bochner-formula-on-an-alexandrov-space-worse-than-on-a-riemannian-mani Why is the Bochner formula on an Alexandrov space worse than on a Riemannian manifold? jiangsaiyin 2013-01-22T18:56:08Z 2013-02-02T15:01:03Z <p>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?</p> http://mathoverflow.net/questions/120109/how-to-compute-the-first-eigenvalue-of-m-r-times-cosh-ts2-left-f How to compute the first eigenvalue of $M = R{ \times _{\cosh t}}{S^2}\left( {\frac{1}{2}} \right)$ jiangsaiyin 2013-01-28T13:41:23Z 2013-01-28T13:41:23Z <p>$$M = R \times N$$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</p> http://mathoverflow.net/questions/119831/how-to-compute-the-first-eigenvalue-of-m-r-times-cosh-tn How to compute the first eigenvalue of $M = R \times {}_{\cosh t}N$ jiangsaiyin 2013-01-25T13:40:09Z 2013-01-27T06:03:12Z <p>$$M = R \times N$$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?</p> http://mathoverflow.net/questions/119444/classification-of-non-compact-riemannian-manifold-with-ric-n-1-and-first-eige classification of non-compact Riemannian manifold with Ric>=-(n-1),and first eigenvalue λ=(n-1)^2/4 jiangsaiyin 2013-01-21T05:59:12Z 2013-01-21T19:13:20Z <p>If M is a non-compact Riemannian manifold with Ric>=-(n-1),we know that the first eigenvalue λ(M)&lt;=(n-1)^2/4.What if λ(M)=(n-1)^2/4,then M would be?Are there any paper on this topic? </p> http://mathoverflow.net/questions/119143/level-set-of-function-f-coincides-with-that-of-f-m-splits level set of function f coincides with that of |▽f|,M splits? jiangsaiyin 2013-01-17T07:42:57Z 2013-01-17T13:38:03Z <p>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?</p> http://mathoverflow.net/questions/118994/examples-of-alexandrov-space-with-sec-1-and-first-eigenvalue-n-12-4 examples of Alexandrov space with sec>=-1 and first eigenvalue =(n-1)^2/4 jiangsaiyin 2013-01-15T16:46:47Z 2013-01-15T16:46:47Z <p>could someone give some examples :nonRiemannian manifold Alexandrov space with sec>=-1 and the first eigenvalue equal to (n-1)^2/4</p> http://mathoverflow.net/questions/110387/about-parabolic-cone about parabolic cone jiangsaiyin 2012-10-23T06:01:16Z 2013-01-02T02:22:00Z <p>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? </p> http://mathoverflow.net/questions/117020/how-many-nonparabolic-ends-guarantee-a-nonconstant-harmonic-function-on-riemannia how many nonparabolic ends guarantee a nonconstant harmonic function on Riemannian manifold? jiangsaiyin 2012-12-22T09:14:59Z 2012-12-24T15:55:10Z <p>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?</p> http://mathoverflow.net/questions/112746/does-this-pde-has-a-general-solution Does this PDE has a general solution? jiangsaiyin 2012-11-18T04:31:43Z 2012-11-18T05:19:41Z <p>$$K\frac{\partial }{{\partial x}}(h\frac{{\partial h}}{{\partial x}}) = \mu \frac{{\partial h}}{{\partial t}}$$ K and u are constants. If no,how to get a asymptotic solution?ie,linearize </p> http://mathoverflow.net/questions/111111/distance-formula-for-warped-product distance formula for warped product jiangsaiyin 2012-11-01T05:23:07Z 2012-11-01T05:23:07Z <p>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.</p> http://mathoverflow.net/questions/110308/is-level-set-of-busemann-function-on-alexandrov-space-again-alexandrov-space Is level set of Busemann function on Alexandrov space again Alexandrov space? jiangsaiyin 2012-10-22T08:03:16Z 2012-10-22T08:03:16Z <p>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? </p> http://mathoverflow.net/questions/109697/laplacian-of-busemann-function-on-hyperbolic-space laplacian of Busemann function on hyperbolic space jiangsaiyin 2012-10-15T08:56:40Z 2012-10-15T15:36:49Z <p>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 $γ : (−\infty, 0] \rightarrow M.$ </p> <p>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 $\Delta b_+,\Delta d_-.$</p> <p>Below is my computation,I don't know whether it's right. Δb+&lt;=limt→+∞(n-1)cotht=n-1 Δb-&lt;=limt→-∞(n-1)cotht=-(n-1) so Δ(b++b−)&lt;=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).</p> http://mathoverflow.net/questions/109716/busemann-function-on-hyperbolic-space Busemann function on Hyperbolic space jiangsaiyin 2012-10-15T13:28:26Z 2012-10-15T15:35:21Z <blockquote> <p><strong>Possible Duplicate:</strong><br> <a href="http://mathoverflow.net/questions/109697/laplacian-of-busemann-function-on-hyperbolic-space" rel="nofollow">laplacian of Busemann function on hyperbolic space</a> </p> </blockquote> <p>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?</p> http://mathoverflow.net/questions/131356/what-does-a-singular-simplex-with-real-coefficient-mean/131358#131358 Comment by jiangsaiyin jiangsaiyin 2013-05-22T10:09:42Z 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 http://mathoverflow.net/questions/131356/what-does-a-singular-simplex-with-real-coefficient-mean/131358#131358 Comment by jiangsaiyin jiangsaiyin 2013-05-22T10:03:19Z 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 http://mathoverflow.net/questions/128307/homotopy-from-a-lipschitz-map-on-the-boundary-to-a-lip-map Comment by jiangsaiyin jiangsaiyin 2013-04-22T06:47:43Z 2013-04-22T06:47:43Z Can you explain it in detail?The difficulty is the homotopy relative to the boundary of E http://mathoverflow.net/questions/127934/only-finitely-many-fundamental-groups-in-mn-k-v-d Comment by jiangsaiyin jiangsaiyin 2013-04-18T12:47:24Z 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? http://mathoverflow.net/questions/127934/only-finitely-many-fundamental-groups-in-mn-k-v-d Comment by jiangsaiyin jiangsaiyin 2013-04-18T11:21:36Z 2013-04-18T11:21:36Z &quot;all the relations are of the form γiγj=γk&quot;means for any γi,γj,there exists a γk such that γiγj=γk? http://mathoverflow.net/questions/127208/left-e-times-0-cup-partial-e-times-i-right-is-a-retract-of-e/127217#127217 Comment by jiangsaiyin jiangsaiyin 2013-04-11T14:21:00Z 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. http://mathoverflow.net/questions/127208/left-e-times-0-cup-partial-e-times-i-right-is-a-retract-of-e Comment by jiangsaiyin jiangsaiyin 2013-04-11T14:19:40Z 2013-04-11T14:19:40Z Oh,I know ,thank you. http://mathoverflow.net/questions/127208/left-e-times-0-cup-partial-e-times-i-right-is-a-retract-of-e Comment by jiangsaiyin jiangsaiyin 2013-04-11T09:43:54Z 2013-04-11T09:43:54Z But where is the middle point of Ex1 projected? http://mathoverflow.net/questions/124487/integral-of-a-harmonic-function-on-a-manifold-with-two-non-parabolic-ends/124497#124497 Comment by jiangsaiyin jiangsaiyin 2013-03-14T12:01:29Z 2013-03-14T12:01:29Z If one parabolic end,one non-parabolic end,the inequality holds? http://mathoverflow.net/questions/122861/must-a-hyperbolic-cone-over-riemannian-manifold-be-manifold Comment by jiangsaiyin jiangsaiyin 2013-02-26T03:17:12Z 2013-02-26T03:17:12Z It must be a manifold,please help me to close the question if you see it. http://mathoverflow.net/questions/122861/must-a-hyperbolic-cone-over-riemannian-manifold-be-manifold Comment by jiangsaiyin jiangsaiyin 2013-02-25T09:28:59Z 2013-02-25T09:28:59Z Sorry,I should write$M = R \times {}_{\cosh \left( t \right)}N$ http://mathoverflow.net/questions/119444/classification-of-non-compact-riemannian-manifold-with-ric-n-1-and-first-eige Comment by jiangsaiyin jiangsaiyin 2013-01-21T06:57:19Z 2013-01-21T06:57:19Z Thank you!I should add &quot;M is non-compact&quot; http://mathoverflow.net/questions/118994/examples-of-alexandrov-space-with-sec-1-and-first-eigenvalue-n-12-4 Comment by jiangsaiyin jiangsaiyin 2013-01-16T02:39:01Z 2013-01-16T02:39:01Z &quot;sharp spectral gap and Li-Yau's estimate on Alexandrov spaces&quot; ,the first page you can see the definition http://mathoverflow.net/questions/117586/harmonic-function-on-alexandrov-spaces Comment by jiangsaiyin jiangsaiyin 2012-12-31T02:07:22Z 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&quot;complete manifolds with positive spectrum.The main obstacle in Alexandrov space is that the bochner formula is not so good as in Riemannian manifold. http://mathoverflow.net/questions/116971/about-the-definiton-of-first-eigenvalue Comment by jiangsaiyin jiangsaiyin 2012-12-21T15:49:05Z 2012-12-21T15:49:05Z I mean why not let the integration equal to 1?