  We construct $p$-adic triple product $L$-functions that interpolate (square
roots of) central critical $L$-values in the balanced region. Thus, our
construction complements that of M. Harris and J. Tilouine.
  There are four central critical regions for the triple product $L$-functions
and two opposite settings, according to the sign of the functional equation. In
the first case, three of these regions are of interpolation, having positive
sign; they are called the unbalanced regions and one gets three $p$% -adic
$L$-functions, one for each region of interpolation (this is the
Harris-Tilouine setting). In the other setting there is only one region of
interpolation, called the balanced region. An especially interesting feature of
our construction is that we get three different $p$-adic triple product $% L
$-functions with the same (balanced) region of interpolation. To the best of
the authors' knowledge, this is the first case where an interpolation problem
is solved on a single critical region by different $p$-adic $L$% -functions at
the same time. This is possible due to the structure of the Euler-like factors
at $p$ arising in the interpolation formulas, the vanishing of which are
related to the dimensions of certain Nekovar period spaces. Our triple product
$p$-adic $L$-functions arise as specializations of $p$-adic period integrals
interpolating normalizations of the local archimedean period integrals. The
latter encode information about classical representation theoretic branching
laws. The main step in our construction of $p$-adic period integrals is showing
that these branching laws vary in a $% p$-adic analytic fashion. This relies
crucially on the Ash-Stevens theory of highest weight representations over
affinoid algebras.

Number Theory