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.