User hapchiu - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T05:19:56Z http://mathoverflow.net/feeds/user/22073 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114493/corona-theorem-in-several-variables Corona Theorem in several variables hapchiu 2012-11-26T07:16:23Z 2013-04-14T08:29:37Z <p>Hallo,</p> <p>I have read about the Corona Theorem (see link:http://en.wikipedia.org/wiki/Corona_theorem). From this one ca deduce that: Let $f_{1}, ..., f_{n}$ be holomorphic bounded functions on the unit disk, which do not all simultaniously vanish. Then there exists bounded holomorphic functions $g_{1}, ..., g_{n}$ such that $\sum_{i=1}^{n}f_{i}g_{i} = 1$. My question is: is this true in several variables? Let say, if the disk is a polydisk or some open, convex ... domain?</p> <p>hapchiu</p> http://mathoverflow.net/questions/124983/sun-structures-on-a-manifold $SU(n)$-structures on a manifold hapchiu 2013-03-19T16:36:41Z 2013-03-19T18:22:47Z <p>I have the following question. Consider a $2n$-dimensional almost complex manifold $M$. Assume that on $M$ there exists a complex valued $n$-form $\Omega$ and a $2$-form $\omega$ such that:</p> <ol> <li><p>$\Omega$ is locally decomposable, i.e. there exists $n$ $1$-forms $\theta_{i}$ such that $\Omega = \theta_{1} \wedge ... \wedge \theta_{n}$.</p></li> <li><p>$\Omega \wedge \omega = 0$.</p></li> <li><p>$|\Omega \wedge \overline{\Omega}| > 0$.</p></li> </ol> <p>How can one show that $M$ admits an $SU(n)$-structure? I think to define charts and show that the transitions functions are elements of $SU(n)$. But how to define the trivialisations ?</p> <p>Greetings hapchiu</p> http://mathoverflow.net/questions/123461/flat-connection-finite-dimensional-space-of-covariant-constant-one-forms Flat connection, finite-dimensional space of covariant constant one forms hapchiu 2013-03-03T06:42:04Z 2013-03-03T08:24:47Z <p>hallo,</p> <p>I have the following question: Let $U\subset \mathbb{R}^{n}$ be an open subset. Furthermore, let $\nabla$ be a flat connection on $U$ (not necessary Levi-Civita). How can one show that the space of covariant constant one forms on $U$ is finite dimensional, i.e. $n$-dimensional real vector space ? is it true ? or no ?</p> <p>hapchiu</p> http://mathoverflow.net/questions/122396/question-in-the-paper-of-robert-bryant-calibrated-embeddings-in-the-special-lagr Question in the paper of Robert Bryant "Calibrated embeddings in the special Lagrangian and coassociative cases" hapchiu 2013-02-20T13:27:02Z 2013-02-20T13:41:49Z <p>Hallo,</p> <p>I am trying to read the paper "Calibrated embeddings in the special Lagrangian and coassociative cases" by Robert Bryant and I have a question. I hope that maybe one of you could give me some answers. The paper can be found here: <a href="http://arxiv.org/abs/math/9912246" rel="nofollow">http://arxiv.org/abs/math/9912246</a> . Here is my question: On page 12, in the middle, there are defined the spaces $\mathfrak{h}_{k}$ = { $x\in \mathbb{R}^{n\times n}$ | $\iota^{*}_{k}(x.\alpha)=0$, $\forall \alpha \in$ $\Lambda^{*}(\mathbb{R}^{n})^{G}$ }. I am not very familiar with the notation. I think $x.\alpha$ means the Lie-derivative of $\alpha$ in the direction $x$. Am I right? If so, using the formula for the Lie-derivative $\mathcal{L}_{X}= \iota d + d\iota$. I do not see how to plug in a matrix $x \in \mathbb{R}^{n\times n}$ in a form defined on $\mathbb{R}^{n}$. How can this be understood? I think, by using the $\mathbb{R}^{n}$-valued form $\nu$ defined by $\nu (v) = u(\pi ' (v))$ , for all $v \in T _{u} F$ (also on page 12 in the middle). But still, for $x \in \mathbb{R}^{n\times n} = Lie(GL(n,\mathbb{R}))=ker(\pi ')$ one gets $\nu(x)=0$. How is this to be understood? I would be very tankfull if somebody could help me with this.</p> <p>greetings hapchiu </p> http://mathoverflow.net/questions/118609/uniqueness-of-kahler-form-with-same-volume Uniqueness of Kähler form with same volume hapchiu 2013-01-11T09:15:15Z 2013-01-22T13:05:41Z <p>Hallo,</p> <p>Let $M$ be a compact real-analytic Riemannian manifold with Riemannian metric $g$. Let $U \subset T^{*}M$ be a open neighbourhood of the zero section. On $U$ there exists a complex structure $J$ and a Kähler form $-i\partial \overline{\partial} \phi$ such that $M$ is a Lagrangian manifold and $\phi = d\phi =0$ on $M$. This shows the existence. Now lets assume that there are two symplectic forms $\omega_{1} = -i\partial \overline{\partial} \phi_{1}$ and $\omega_{2} = -i\partial \overline{\partial} \phi_{2}$ in the same complex structure and such that $M$ is a Lagrangian manifold with respect to both and $\omega_{1}^{n} = \omega_{2}^{n}$ (they have the same volume), where $n$ is the dimension of $M$. My question is: are these Kähler forms the same? If not, what assumptions does one need to make in order that they are equal? Or, is there any chance, at all, to make them equal?</p> <p>hapchiu</p> http://mathoverflow.net/questions/117843/solution-of-a-pde-and-its-uniqueness Solution of a PDE and its uniqueness hapchiu 2013-01-02T06:49:07Z 2013-01-03T13:09:44Z <p>Hallo,</p> <p>consider $f: U \times I \rightarrow \mathbb{R}$, where $U \subset \mathbb{R}^{n}$ and $0 \in I \subset \mathbb{R}$ be two open sets. I am looking for the solution $f$ of the following PDE $\sum_{i=0}^{n} (\frac{\partial^{2}f}{\partial t^{2}})^{i} K_{i}(x,t,f,\frac{\partial f}{\partial t}, \frac{\partial^{2} f}{\partial t \partial x_{j}}, \frac{\partial^{2} f}{\partial x_{k} \partial x_{j}}) = 0$, with initial condition $f(x,0) = \frac{\partial f}{\partial t}(x,0) = 0$. Here the $K_{i}$ are analytic functions depending of several variables and $f$ should also be analytic. The solution should not be on the whole $U \times I$, maybe on some smaller open set. Does there exists a solution? If yes, is this solution unique? When does a solution exists? I have read what bryant wrote and it seems now clearly to me. But I have a different suggestion (I am not 100 % convinced but maybe one can point out where I did a mistake). Here is what I thaught: Differentiate the whole PDE expression with respect to $t$, then one obtains: $\frac{\partial }{\partial t}(\sum_{i=0}^{n} (\frac{\partial^{2}f}{\partial t^{2}})^{i} K_{i}(x,t,f,\frac{\partial f}{\partial t}, \frac{\partial^{2} f}{\partial t \partial x_{j}}, \frac{\partial^{2} f}{\partial x_{k} \partial x_{j}})) = 0$. Thus one obtains $\sum_{i=1}^{n}i(\frac{\partial^{2}f}{\partial t^{2}})^{i-1} \cdot (\frac{\partial^{3}f}{\partial t^{3}})K_{i} + (\frac{\partial^{2}f}{\partial t^{2}}) \cdot \frac{\partial K_{i}}{\partial t} + (\frac{\partial^{2}f}{\partial t^{2}}) \cdot \frac{\partial K_{0}}{\partial t} = 0$. Now we write the PDE as: $(\frac{\partial^{3}f}{\partial t^{3}}) \cdot F + G = 0$, where $F = \sum_{i=1}^{n}i(\frac{\partial^{2}f}{\partial t^{2}})^{i-1}K_{i}$ and $G = \sum_{i=1}^{n}(\frac{\partial^{2}f}{\partial t^{2}}) \cdot \frac{\partial K_{i}}{\partial t} + (\frac{\partial^{2}f}{\partial t^{2}}) \cdot \frac{\partial K_{0}}{\partial t}$. Now if $F$ does not vanish on ${t=0}$ we consider the equation $(\frac{\partial^{3}f}{\partial t^{3}}) \cdot F + G = 0$ and can use Cauchy-Kovalewskaya with appropriate initial conditions. This was my guess. Is this right? Where is the mistake? </p> <p>hapchiu</p> http://mathoverflow.net/questions/115692/isometric-embedding-of-a-neighbourhood-of-a-totally-real-submanifold-in-a-kahler Isometric embedding of a neighbourhood of a totally real submanifold in a Kähler manifold hapchiu 2012-12-07T08:50:44Z 2012-12-07T16:03:51Z <p>Hallo,</p> <p>Let $(M,J,\omega)$ be a real-analytic Kähler manifold. Let furthermore $A \subset M$ be a real analytic, totally real, Lagrangian submanifold and set $g := h|_{A}$. Where $h$ is the Kähler metric on $M$. $g$ is now a Riemannian metric on $A$. Let $U$ be an arbitrary small neigbourhood of $A$ in $M$. Is it possible to embedd $U$ in some $\mathbb{C}^{N}$ isometrically? I think its always possible to embed such an arbitrary small neighbourhood $U$ in $\mathbb{C}^{N}$ for some $N$. But can this be also done isometrically?</p> <p>hapchiu</p> http://mathoverflow.net/questions/115583/real-analytic-submanifolds-of-mathbbrn Real analytic submanifolds of $\mathbb{R}^{n}$ hapchiu 2012-12-06T07:02:02Z 2012-12-06T22:58:06Z <p>Hallo,</p> <p>Let $(M,g)$ be a Riemannian $k$-dim real analytic submanifold of $\mathbb{R}^{n}$. Is it true that $M$ in $\mathbb{R}^{n}$ looks locally (in a small neigbourhood around some point in $M$) as the zero set of some polynomials? If yes, why ? Are there any references? </p> http://mathoverflow.net/questions/115420/constructing-the-imaginary-part-of-a-holomorphic-function Constructing the imaginary part of a holomorphic function hapchiu 2012-12-04T16:57:55Z 2012-12-04T18:09:30Z <p>Hallo,</p> <p>Let $f: U \rightarrow \mathbb{R}$ be a analytic function, where $U \subset \mathbb{C}^{n}$ is a open set (paracompact, starshaped or convex i.e. sufficiently nice). Does there exist a function $\varphi : U \rightarrow \mathbb{R}$ such that $f + i \varphi$ is holomorphic? Is this possible?</p> <p>hapchiu </p> http://mathoverflow.net/questions/115092/kahler-form-on-complex-lie-group Kähler form on complex Lie group hapchiu 2012-12-01T17:48:09Z 2012-12-01T19:48:40Z <p>Hallo,</p> <p>Let $G$ be a semi-simple, compact Lie Group. Consider its complexification $G_{\mathbb{C}}$. Does there exist a Kähler structure on $G_{\mathbb{C}}$ which is $G$-invariant (maybe in a neighbourhood of $G$ in $G_{\mathbb{C}}$)?</p> <p>hapchiu</p> http://mathoverflow.net/questions/114849/isometric-embedding-of-a-real-analytic-riemannian-manifold-in-a-compact-kahler-ma Isometric embedding of a real-analytic Riemannian manifold in a compact Kähler manifold hapchiu 2012-11-29T06:35:58Z 2012-11-29T08:36:40Z <p>Hallo,</p> <p>It is a known fact that any real-analytic Riemannian manifold $M$ admits a isometric embedding in a Kähler manifold $\Omega$, where $M$ is totally real in $\Omega$. Of $\Omega$ can be taught of as some open neighbourhood of the zero section of the cotangent bundle $T^{*}M$. This complex manifold is far from being compact. My question is now: Can one choose the complex manifold $\Omega$ not to be just some open neighbourhood of $M$ but to be a compact Kähler manifold? In other words: Can one embed any real-analytic compact Riemannian manifold isometrically in a compact Kähler manifold, such that the real-analytic Riemannian manifold is totally real in the Kähler manifold? Are there any references on this topic? What furter assumptions does one need in order that this works? Or is it entirely impossible? If so, why? By compact I mean compact without boundary!</p> <p>hapchiu</p> http://mathoverflow.net/questions/112689/analytic-lagrangian-submanifolds Analytic Lagrangian Submanifolds hapchiu 2012-11-17T15:24:22Z 2012-11-26T09:32:51Z <p>Hallo,</p> <p>I am looking for a preprint "Analytic Lagrangian Submanifolds" by Guillemin, Sternberg. I googled it but without any success. Does any one know how I could get this preprint. Or are there similar ones? I am actually interested in understanding better the construction of a defining phase function for a Lagrangian submanifold and to understand the uniqueness. Is there any other literature on that? Actually I am interested in the proof of the following: Let $M$ be a connected Lagrangian submanifold of a Kähler manifold $\Omega$, then there is a neighbourhood $U$ of $M$ in $\Omega$ and a unique defining phase function $\phi$ on $U$ for $M$. </p> <p>hapchiu </p> http://mathoverflow.net/questions/112281/polarisation-in-a-nighbourhood-of-a-lagrangian-submanifold Polarisation in a nighbourhood of a Lagrangian submanifold hapchiu 2012-11-13T13:31:44Z 2012-11-17T21:43:11Z <p>Hallo,</p> <p>Let $(X, \omega)$ be a symplectic manifold of dimension $2n$ and $\omega$ is an exact symplectic form i.e. $\omega = -d\alpha$. Let furthermore $M \subset X$ be a Lagrangian submanifold such that $\alpha = 0$ of $TX|_{M}$. I am interested in the following questions:</p> <ol> <li><p>Is there a unique polarisation defined on $X$ near $M$ which is transversal to $M$ and whose one form is $\alpha$ ? By polarisation I mean the following: A polarisation of a symplectic manifold $X$, with symplectic form $\omega$, is a smooth assignment of a Lagrangian subspace of $T_{x}X$ to each $x \in X$ in such a way that this assignment is integrable.</p></li> <li><p>If 1. is true, is there a symplectic diffeomorpism $\Phi$ of a neigbourhood of $M$ is $X$ with a neigbourhood of $M$ in its cotangent bundle which carries the leaves of the polarisation into the standard cotangent fibration of $T^{*}M$ ?</p></li> </ol> <p>Actually I know that these results are true. I would like to see the proof of them. Are there any references where I can look them up? If so, can you please tell me where these references can be found? Thanks a lot!</p> <p>hapchiu</p> http://mathoverflow.net/questions/112078/harmonic-function Harmonic Function? hapchiu 2012-11-11T15:05:28Z 2012-11-11T15:05:28Z <p>Hi,</p> <p>Let $\varphi : U \rightarrow \mathbb{C}$ be a holomorphic function, where $U \subset \mathbb{C}^{n}$ containing $0$. Is the function $u(x_{1}, ..., x_{n}, y_{1}, ..., y_{n}) := Im( \int_{0}^{(x_{1} + iy_{1}, ..., x_{n} + iy_{n})} \sqrt{\varphi (\xi)} d \xi)$ harmonic. Where integration should be understood: integrate along any path from $0$ to $(x_{1} + iy_{1}, ..., x_{n} + iy_{n})$ (by Cauchy the integral is path independet). With harmonic I mean: $\Delta u = \frac{\partial^{2}u}{\partial x_{1}^{2}} + ... + \frac{\partial^{2}u}{\partial x_{n}^{2}} + \frac{\partial^{2}u}{\partial y_{1}^{2}} + ... + \frac{\partial^{2}u}{\partial y_{1}^{n}} = 0$. I was reading about this and in 2 dimensions it is harmonic. Now I am actually interested if its also harmonic in more dimensions. Is it? </p> <p>hapchiu</p> http://mathoverflow.net/questions/111618/rotation-in-hyperkahler-manifolds Rotation in Hyperkähler manifolds hapchiu 2012-11-06T05:10:23Z 2012-11-07T21:46:49Z <p>Any Hyperkähler manifold has 3 complex structures $I_{1}, I_{2}, I_{3}$. Assume that there is an additional complex structure $J$. Can this be written as $J = aI_{1} + bI_{2} + cI_{3}$, where $(a,b,c) \in S^{2} \subset \mathbb{R}^{3}$? Hope the question is not too trivial :).</p> http://mathoverflow.net/questions/90805/example-of-special-lagrangian-submanifold/90973#90973 Answer by hapchiu for example of special lagrangian submanifold hapchiu 2012-03-12T09:49:10Z 2012-03-12T09:49:10Z <p>I think, eaven in this case you mentioned there counterexamples, but cannot be given explicitely. See the post of Bryant (above), part 1. Am I right?</p> <p>hapchiu</p> http://mathoverflow.net/questions/123461/flat-connection-finite-dimensional-space-of-covariant-constant-one-forms Comment by hapchiu hapchiu 2013-03-03T07:51:33Z 2013-03-03T07:51:33Z anyone has an idea ? http://mathoverflow.net/questions/122396/question-in-the-paper-of-robert-bryant-calibrated-embeddings-in-the-special-lagr Comment by hapchiu hapchiu 2013-02-20T14:00:31Z 2013-02-20T14:00:31Z is there any interpretation of this using the Lie-derivative ? http://mathoverflow.net/questions/122396/question-in-the-paper-of-robert-bryant-calibrated-embeddings-in-the-special-lagr Comment by hapchiu hapchiu 2013-02-20T13:43:04Z 2013-02-20T13:43:04Z i have edited my question once again so one actually can read it :). http://mathoverflow.net/questions/118609/uniqueness-of-kahler-form-with-same-volume Comment by hapchiu hapchiu 2013-01-12T09:14:14Z 2013-01-12T09:14:14Z @Bryant: I take any complex structure, since by Bruhat-Whitney in a neighbourhood of the zero section they are biholomorphic. http://mathoverflow.net/questions/117843/solution-of-a-pde-and-its-uniqueness Comment by hapchiu hapchiu 2013-01-02T12:38:04Z 2013-01-02T12:38:04Z what do you mean by characteristic? http://mathoverflow.net/questions/117843/solution-of-a-pde-and-its-uniqueness Comment by hapchiu hapchiu 2013-01-02T09:13:11Z 2013-01-02T09:13:11Z why do you think so? http://mathoverflow.net/questions/115692/isometric-embedding-of-a-neighbourhood-of-a-totally-real-submanifold-in-a-kahler Comment by hapchiu hapchiu 2012-12-07T13:29:07Z 2012-12-07T13:29:07Z I thaught that one can do the following: since $A$ is real analytic we can use the real analytic version of Nash embedding theorem and consider $A$ as a real analytic Riemannian submanifold of some $\mathbb{R}^{N}$. Then locally $A$ is the zero set of some real analytic functions. Extend these functions holomorphically and then patch them together, since on the overlaps of some open sets in $A$ these real analytic functions are the same, and the extension of them would be the same holomorphic function. Is this possible? http://mathoverflow.net/questions/115692/isometric-embedding-of-a-neighbourhood-of-a-totally-real-submanifold-in-a-kahler Comment by hapchiu hapchiu 2012-12-07T13:18:53Z 2012-12-07T13:18:53Z yes, is it possible if the embedding is holomorphic? http://mathoverflow.net/questions/115420/constructing-the-imaginary-part-of-a-holomorphic-function/115423#115423 Comment by hapchiu hapchiu 2012-12-04T17:28:00Z 2012-12-04T17:28:00Z in dimension $\geq 2$? does the same still hold? http://mathoverflow.net/questions/115092/kahler-form-on-complex-lie-group/115103#115103 Comment by hapchiu hapchiu 2012-12-02T08:59:56Z 2012-12-02T08:59:56Z what do you mean by average its pullbacks under left and right multiplications if $G$ is compact??? http://mathoverflow.net/questions/115092/kahler-form-on-complex-lie-group/115103#115103 Comment by hapchiu hapchiu 2012-12-02T08:27:31Z 2012-12-02T08:27:31Z Is there any reference where I can see how the proof is done? http://mathoverflow.net/questions/114849/isometric-embedding-of-a-real-analytic-riemannian-manifold-in-a-compact-kahler-ma/114859#114859 Comment by hapchiu hapchiu 2012-11-29T09:27:42Z 2012-11-29T09:27:42Z If all this works is the resulting complex manifold then compact (compact without boundary)? http://mathoverflow.net/questions/114849/isometric-embedding-of-a-real-analytic-riemannian-manifold-in-a-compact-kahler-ma/114859#114859 Comment by hapchiu hapchiu 2012-11-29T09:12:49Z 2012-11-29T09:12:49Z Thanks for the answer Dmitri. Is there any reference on this? http://mathoverflow.net/questions/114493/corona-theorem-in-several-variables Comment by hapchiu hapchiu 2012-11-27T11:33:23Z 2012-11-27T11:33:23Z why does this paper not solve the posted problem? http://mathoverflow.net/questions/114493/corona-theorem-in-several-variables Comment by hapchiu hapchiu 2012-11-27T11:26:15Z 2012-11-27T11:26:15Z what do you mean by &quot;BMO&quot;?