  We have examined UV spectra recorded by the Space Telescope Imaging
Spectrograph (STIS) on the Hubble Space Telescope for three stars, HD32309, 41
Ari, and $\eta$~Tel, that are located well inside the boundary of the Local Hot
Bubble in our search for absorption features of Si IV, C IV, and N V that could
reveal the presence of an interface between the local warm ($T\sim 7000$ K)
neutral medium and a more distant hot ($T\sim 10^6$ K) interstellar medium. In
all cases, we failed to detect such ions. Our most meaningful upper limit is
that for log N(C IV)< 11.86 toward HD32309, which is below the expectation for
a sight line that penetrates either a conductive/evaporative interface or a
turbulent mixing layer. We offer conjectures on the reasons for these negative
results in terms of either a suppression of a conductive layer caused by the
shielding of the local cloud by other clouds, which may make it more difficult
for us to sense discrete absorption features from gases at intermediate
temperatures, or by the presence of a tangential magnetic field at most
locations on the surface of the local cloud.
