k-hessian equation and k-curvature equation

here is the problem, how to understand k-hessian equation and k-curvature equation.

k-hessian equation

k-hessian equation is:

$latex H_k(u)=\sigma_k(D^2(u))=f$ (*)

where u is admissible, i.e. $latex \forall 1\leq i\leq k$, $latex \sigma_i(D^2(u))\geq 0$. this is just the condition to make (*) be a elliptic equation.

The most important result is the following three:

1.sovable (*) with direchlet boundary condition.

This is mainly the contribution of Caffaralli in 90’s. According flexible function and maximum principle we can establish the $latex C^{1,\alpha}$ estimate and $latex C^{2,\alpha}$ estimate in the inter. And the $latex C^{2,\alpha}$ estimate near the boundary is establish according to the conformation invariant and some perbutation of the solution of k-hessian equation after special rescaling.

2.Hessian measure.

This is mainly the work of X.J.Wang and Trudinger. they proved:

in the meaning of viscosity solution, if $latex \sigma_k(D^2(u))=f$. then we can associate a measure $latex \mu$ with $latex u$,and the following is right:

when $latex u\in C^2(\Omega)$, $latex \mu(B_r(x))=\int_{B_r{x}}\sigma_k(D^2(u))$.

if $latex u_1,..,u_n,…$ coverage to $latex u$. then $latex \mu_1,…,\mu_n,…$ coverage to $latex \mu$ in weak sense.


this is merely depend on a priori estimate on $latex u$

3.pointwise estimate corresponding wolff potential.

the Wolff potential is:

$latex W^{\mu}_{k}(x,r)= \int_{0}^r(\frac{\mu(B_t(x))}{t^{n-2k}})^{\frac{1}{k}}\frac{1}{t}dt$

We can easily use rescaling to understand the reasonable of this potential, and use this potential Lubutin establish the following pointwise estimate:

$latex u\in \Phi_k(B_{4R}(x))$, $latex u\leq 0$, then we have:

$latex W^{\mu}_k(x,\frac{R}{2})\leq |u(0)| \leq W^{\mu}_k(x,2R)-sup_{B_{2R}}|u|$


the RHS could look as a corollary of classical A-B-P estimate. the LHS need combine several observation. mean-value property and some else.

This result could use to establish some result on singularity point can be removable.

k-curvature equation

1.sovable (*) with direchlet boundary condition.

This is also established by cafferalli.

2.curvature measure.

This is established very recently. mean curvature equation in 2014, by perron lift and modified, general case in 2016 by more complex calculate and method.

3.pointwise estimate corresponding wolff potential.

This still do not established, and is the main thing I focus on. Due to we can look as k-curvature as a “projection” of k-hessian equation, Calderon-Zegmund decomposition and the estimate of k-hessian equation maybe useful.


 My ideas

look is as “average” of “loop space”, “surface space”.

1.Grassmannian bundle

n algebraic geometry, the Grassmann d-plane bundle of a vector bundle E on an algebraic scheme X is a scheme over X:
$latex {\displaystyle p:G_{d}(E)\to X}$
such that the fiber

$latex {\displaystyle p^{-1}(x)=G_{d}(E_{x})}$ is the Grassmannian of the d-dimensional vector subspaces of $latex E_x$. For example,

$latex {\displaystyle G_{1}(E)=\mathbb {P} (E)}$ is the projective bundle of E. In the other direction, a Grassmann bundle is a special case of a (partial) flag bundle. Concretely, the Grassmann bundle can be constructed as a Quot scheme.

Like the usual Grassmannian, the Grassmann bundle comes with natural vector bundles on it; namely, there are universal or tautological subbundle S and universal quotient bundle Q that fit into

$latex {\displaystyle 0\to S\to p^{*}E\to Q\to 0}$.
Specifically, if V is in the fiber p−1(x), then the fiber of S over V is V itself; thus, S has rank $latex r = rk(E)$ and

$latex {\displaystyle \wedge ^{r}S}$ is the determinant line bundle. Now, by the universal property of a projective bundle, the injection

$latex {\displaystyle \wedge ^{r}S\to p^{*}(\wedge ^{r}E)}$ corresponds to the morphism over X:
$latex {\displaystyle G_{d}(E)\to \mathbb {P} (\wedge ^{r}E)}$,
which is nothing but a family of Plücker embeddings.

The relative tangent bundle $latex T Gd(E)/X$ of $latex Gd(E)$ is given by[1]
$latex {\displaystyle T_{G_{d}(E)/X}=\operatorname {Hom} (S,Q)=S^{\vee }\otimes Q,}$
which is morally given by the second fundamental form. In particular, when d = 1, the early exact sequence tensored with the dual of S = O(-1) gives:
$latex {\displaystyle 0\to {\mathcal {O}}_{\mathbb {P} (E)}\to p^{*}E\otimes {\mathcal {O}}_{\mathbb {P} (E)}(1)\to T_{\mathbb {P} (E)/X}\to 0}$,
which is the relative version of the Euler sequence.

2.Explain of the fully nonlinear elliptic equation

Now, we could consider the determination $latex \sum_{i_1,…,i_k\in\{1,…,n\}}det(u_{ij})_{i,j\in \{i_1,…,i_k\}\times\{i_1,…,i_k\}}$ as the determination of transform: $latex (u_{i_1},…,u_{i_k}) \longrightarrow (e_{i_1},…,e_{i_k})$.

Now we need to understand $latex \sigma_k(D^2(u))=f$ at a point $latex x_0$ as the average of determination of transform matrix of $latex (u_{i_1},…,u_{i_k}) \longrightarrow (e_{i_1},…,e_{i_k})$ on Grassmannian manifold $latex G_k(x_0)$ is equal to $latex f(x_0)$, i.e.:

$latex \int_{G_k(x_0)} det(\frac{\partial u_{i_a}}{\partial e_{i_b}})     d\mu=f(x_0)$

where $latex \mu$ is the natural haar measure on $latex G_k(x_0) \simeq G_k$.

But the difficult to make the argument rigorous is that $u_i$ is scale and $e_i$ is vector.