User the common crane - MathOverflow

Decomposing bilinear forms in Hilbert spaces

2013-01-12T08:02:35Z

You are given a complex Hilbert space $H$ with two equivalent Hilbert space structures $<,>$ and $<,>'$. Define $<,>''=<,> + <,>'$ to be the sum of our two scalar products. Suppose you are given a complex symmetric bilinear form $B: H\times H \to \Bbb C$, with norm $c$ over $<,>''$ i.e. $$B(x,x) \leq c(|x|^2 + |x|'^2) , \ x \in H. $$ Does there exist complex symmetric bilinear forms $A$ and $C$, such that $B=A+C$ and $$A(x,x) \leq c|x|^2 , \ x \in H, $$ $$C(x,x) \leq c|x|'^2 , \ x \in H. $$ It is clear that if $A$ and $C$ have norm less then $c$ with respect to the respective norms, then $B$ has norm less then $c$ with respect to the "sum" norm. My questions is, do all operatods $B$ arise this way?

Probably not. However I keep thinking about this on and off, and as embarassing as it is, I can't come up with a counterexample even in the $\Bbb C^n$ case.

Morse lemma with least amount of regularity.

2013-01-02T23:28:08Z

I recently came across with $C^2$ Morse functions in my work and as I was reviewing some of the stuff I learned about Morse theory, I noticed that all the proofs of the Morse lemma I could come across with work only for $C^3$ Morse functions.

A Google search was inconclusive about the existence of a Morse lemma for Morse functions $f: M \to \Bbb R$ with lesser regularity then $C^3$, where $M$ is a smooth finite dimensional manifold.

A reference is perhaps the best possible answer, but any chunk of information will be appreciated.

variation of the obstacle in the obstacle problem

2012-12-22T15:32:45Z

Suppose $D \subset \Bbb C$ with smooth boundary. Let $f \in C^{1,1}(D)$. Let $\varphi$ be the supremum of all members in the set

$$\lbrace g \in C^{\infty}(\overline{D})| g \ is \ subharmonic \ and \ g \leq f\rbrace$$

It is known that $\varphi$ is subharmonic in $D$ and harmonic on $E =\lbrace \varphi < f\rbrace$, the complement of the contact set. It is also known that $\varphi$ is C^{1,1}.

Suppose now that $\lbrace f_t \rbrace$, $t \in(0,1)$ is a smooth family of smooth obstacles. Will the variation of the family $\lbrace \varphi_t \rbrace$ be at least $C^1$?

If this is too much to ask, then let $\tilde E \subset (0,1)\times D$, be the open? domain for which each $t-$slice $E_t$ is the set where $\varphi_t$ will be harmonic. Will the variation of $\lbrace \varphi_t\rbrace$ be $C^1$ on $\tilde E$?

Any reference or chunk of information will be appreciated.

I don't know almost anything about obstacle problems (but you know this by now :D). There seems to be no obvious reference out there. Can someone recommend something to a grad student who is faimiliar with the Gilbarg and Trudinger stuff but not much more?

Deformation of Lagrangian manifolds

2012-06-01T15:38:03Z

I read recently that on a symplectic manifold $M$, the infinitesimal deformations of a Lagrangian manifold $L$ can be identified with closed 1 forms in $T^*L$ (cotangent bundle of L).

How can this correspondance be made? I suppose that one somehow has to use Weinstein's tubular neighborhood theorem, but I can't write down the required map.

I am sure that this construction is standard in sympletic geometry so if someone knows a good reference please let me know.

Subharmonic envelope

2012-11-07T05:55:19Z

I came across a more complicated version of the following problem. It is so elementary, I think that there had to be some research done on this in the past. If someone has any ideas please let me know.

Suppose $u: \bar D \to \Bbb R$ is a smooth function on the closure of the unit disc in $\Bbb C$. We introduce the following family: $$S_u = \lbrace v: D \to \Bbb R \mid v \leq u, v \text{ is continuous and subharmonic }\rbrace$$ Let $h$ be the upper envelope of the supremum of all elements in $S_u$. Then $h$ is an usc subharmonic function.

The question is as follows. Under what conditions on $u$ is the difference $\Delta = u(0)-h(0) \geq 0$ zero. This quantity is zero when $u$ is subharmonic, but its fate seems to be unclear in other situations.

Same betti numbers as $\Bbb{CP}^n$

2012-07-01T06:01:27Z

I am sure that there is an answer out there for the following question. If one is given an n dimensional Kahler manifold $X$ with Betti numbers that are the same as in the case of $\Bbb{CP}^n$, then is $X$ holomorphically diffeomorphic to $\Bbb{CP}^n$? This is of course true in the one dimensional case, but other then that I am clueless. If this is textbook stuff that I might have missed, references will be appreciated :).

Reference request: parabolic PDE

2012-10-07T05:42:02Z

I want to learn about parabolic PDE and it seems to me that there is no established reference as far as where one should look if one wants to learn the subject from basics.

I think I have a firm grip on elliptic PDE after going through the first part of Gilbarg and Trudinger + some Monge-Ampere stuff. But that concludes my PDE background at this moment.

Can someone provide me with a good textbook for parabolic PDE? Any chunk of information will be appreciated.

EDIT: I would like to learn about parabolic PDE arising in geometry, mostly the Kahler-Ricci flow and related questions. But since I am new to this approach perhaps a more broad introduction would be appropriate.

Answer by The Common Crane for Hopf Boundary Point Lemma

2012-10-03T19:51:00Z

I don't know how late this is for an answer, but as long as at point $x_0$ of $\partial \Omega$ where the assumptions of the Hopf Boundary lemma are satisfied, $l(x_0)$ is not tangential to the boundary(outgoing) then the bound $\partial u/ \partial l(x_0) >0$ will be satisfied. Infact you don't even need smoothness of the boundary, as it is detailed in page 34 of Gilbarg and Trudinger.

Frobenius theorem with lesser regularity

2012-09-04T22:40:30Z

Clearly, if one is given a $C^1$ sub-bundle $V$ of the tangent space of a smooth manifold $M$, wheather $V$ comes from a $C^2$ foliation of the manifold is decided by the conditions of the Frobenius theorem.

Of course one can define a $C^1$ foliation of a $C^0$ sub-bundle $V$ as well. However the conditions of the Frobenius theorem are not applicable anymore since $V$ is not differentiable hence there is no good? notion of Lie product.

Does anyone know of an analog of the Frobenius theorem for $C^0$ sub-bundles, or any result in this direction?

Special Morse function on a Riemann surface

2012-07-13T21:16:00Z

Let $f: S \to \Bbb R$ be a Morse function on a Riemann surface. Let $x_0$ be a saddle point of $f$. Since $x_0$ is a critical point of $f$, it makes sense to talk about the bilinear forms $f_{z\overline{z}}(x_0)$, $f_{zz}(x_0)$ and $f_{\overline{z}\overline{z}}(x_0)$ as components of the real Hessian of $f$ at $x_0$ (if one only allows only complex coordinates).

Can $f$ be arranged so that the Hermitian form $f_{z\overline{z}}$, which can be defined on all $S$, is arbitrarily small (compared to some fixed Hermitian metric) and $|f_{zz}(x_0)|$(as a number) is arbitrarily big in a fixed coordinate system around $x_0$?

Perturbation of Morse function at a critical point

2012-07-12T16:45:49Z

I recently learned from a knowledgeable person that for a Morse function $f: M \to R$ with a critical point $x_0$, one can perturb $f$ in such a fashion that the new function has the same critical points and the Hessian at $x_0$ can be arbitrarily arranged if the index is unchanged.

I have managed to prove this result, but it would be nice to know a textbook where this result can be found, since I need to refer to it. I think people who know more Morse theory than I do can help out.

Perturbation of Morse function

2012-07-01T15:56:21Z

The following question is probably classic in Morse theory, so a reference to an existing result should be sufficient. I don't know much about Morse theory and I am dealing with the following situation. I have a compact manifold $X$ and I have a Morse function $f$ on it with a saddle point at $x_0$. I don't like the properties of $f$(for some "mysterious" reason) so I take a parametrization around $x_0$, taking $x_0$ to the origin and containing no additional critical points, and add to $f$ a function of the type $\epsilon \rho Q$, where $Q$ is a quadratic polynomial, $\epsilon$ is a small number and $\rho$ is a bump function living in the coordinate patch that is identically equal to 1 in a neighborhood of the origin.

Clearly this new function still has a critical point at $x_0$. My question is: for small enough $\epsilon$ is this new function still Morse, with the same critical points as the original function $f$?

Curvature and Symmetry on Kahler manifolds

2012-03-12T04:21:33Z

Hi there,

Suppose $X$ is a Kahler manifold that has an analytic isometry $S$, with $S^k = Id$ ($k \in \Bbb N$). In a situation like this(maybe with additional assumptions on $X$) can one say something about the positivity/negativity of the curvature of $X$? Particularly I would be interested in instances where the bisectional curvature might be positive/negative.

If this was already studied(it might be possible since I am relatively new to the field). Can someone please provide some references?

vector valued BVP for ODE's

2012-01-14T11:48:35Z

I am dealing with a vector valued second order homogeneous BVP:

$\ddot u(t) = A(t)\dot u(t) + B(t)u(t)$ with $u(0)=u(1)=0.$

where $A$ and $B$ are $n \times n$ matrices with smooth coefficients and $u$ is a vector valued smooth function on $[0,1]$.

Are there any results that say something about the non-trivial solutions of this equation(their existence, their number, a priori estimates) without too "stringent" conditions on $A$ and $B$. Or is there any criteria implying the non-existence of non-trivial solutions?.

Any help or reference is highly appreciated.

Strong minimum principle for maximal plurisubharmonic functions

2012-01-03T18:55:57Z

Suppose $u$ is a bounded maximal plurisubharmonic function in a bounded domain $D \in \Bbb C^n$. If $u$ is $C^2$ one can see that $u$ cannot have a local strict minimum inside $D$. Is there an analog of this result, when $u$ is not $C^2$-smooth? Any counterexamples?

Symbol of pseudodiff operator

2011-09-20T16:53:04Z

Hello,

I am trying to understand the calculus of pseudodifferential operators on manifolds. All the textbooks I could put my hand on define the principal symbol of a pseudodifferential operator locally, then prove that it transforms well, hence becomes a "global" object. Is there any good way to define the the principal symbol without coordinate patches? Am I asking too much here :)?

Monge Ampere equations

2011-12-16T03:36:01Z

I am a graduate student trying to understand complex Monge-Ampere equations(mostly on complex manifolds with or without boundary, but also in C^n), but I can't put my hand on any monograph/textbook discussing this problem thoroughly. Is there anything out there that could help me? If there isn't, can any of you folks tell me with what articles I should start my reading?

Any piece of information is appreciated.

Comment by The Common Crane

2013-01-04T04:44:12Z

thank you Sir !

Comment by The Common Crane

2013-01-02T22:50:28Z

Thank you for your careful answer.

Comment by The Common Crane

2012-12-22T22:13:41Z

Indeed. Nice observation. My intention with the first question was to lead into the second one which seems more involved. I think in general only $C^{0,1}$ variation is possible. As the boundary of the contact set varies it is hard to believe that along this set full $C^1$ variation is possible(as your example also suggests).

Comment by The Common Crane

2012-11-07T06:16:23Z

Sorry for getting back so late. I think you still don't completely understand me, I want to minimize $f_{z\bar z}$ not just at $x_0$ but all over $S$. What you wrote was very clear for me, and it shows that you can do this at $x_0$. But to minimze $f_{z\bar z}$ one has to be more careful. Thanks for your willingness to help, I will accept your answer.

Comment by The Common Crane

2012-10-10T00:20:31Z

Somebody knowledgeable recommended Lieberman's book because it is similarly written to Gilbarg & Trudinger. Is it worth it compared to other choices?

Comment by The Common Crane

2012-10-07T20:53:59Z

Thanks for the suggestions. As it was rightfully asked, I edited my question to reflect my motivation of study. Perhaps this narrows a few things down. Does it?

Comment by The Common Crane

2012-09-05T20:31:32Z

I just found the following on MO that is basically the same question: mathoverflow.net/questions/12266/… Has I seen that question I would have not asked this one. Thanks for the answers though.

Comment by The Common Crane

2012-07-22T11:53:27Z

You are right again. Thanks for your insistance. But I want is that $f_{z\overline{z}}$ is arbitrarily small all over $S$ and $f_{zz}$ is arbitrarily big at $x_0$ only.

Comment by The Common Crane

2012-07-18T21:54:15Z

Thanks for the response, but you got me wrong on this one I think. I was asking if $f$ can be arranged, not if the coordinates can be arranged ... if I am not mistaken, you proved here that the elements of the Hessian transform well.

Comment by The Common Crane

2012-07-07T05:48:36Z

intersting. Thanks!

Comment by The Common Crane

2012-07-01T18:04:16Z

Thanks again. Although if one uses the Morse lemma for $f$ in a small neighborhood this is just as clear. I am learning Morse theory on the fly :)

Comment by The Common Crane

2012-07-01T17:25:17Z

"But in such neighborhoods the only critical points are the critical points of the original function f, at least if $\epsilon_n$ is small enough". Why is this obvious?

Comment by The Common Crane

2012-07-01T17:23:31Z

I think I declared victory too soon :(. How does it follow that for small $\epsilon$ the new function will have the same critical points as the original $f$.

Comment by The Common Crane

2012-07-01T17:12:14Z

No. I am dealing with quantities that are coordinate independent!

Comment by The Common Crane

2012-07-01T16:52:41Z

thanks. A reference is just what I wanted.